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Nonuniform Space-Time Codes for Layered Source Coding


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T ABLE OF CONTENTS page ACKNOWLEDGMENTS............................ iv LISTOFFIGURES............................... vii ABSTRACT................................... ix CHAPTER 1INTRODUCTION............................. 1 2BROADCASTCHANNELS........................ 5 3LAYEREDSOURCECODING...................... 6 4SPACE-TIMECODING.......................... 9 4.1ErrorPerformanceofSTBCSystems............... 9 4.2DierentialModulationforMIMOSystems............ 18 4.3NonuniformSpace-TimeCodes................... 21 4.3.1DesignCriteria........................ 22 4.3.2UnionBoundontheErrorProbability........... 23 4.4DesignExamples.......................... 24 4.4.1 R b =1, R a =1Codefor2TXAntennas......... 24 4.4.2 R b =2, R a =1Code.................... 30 4.4.3ReceiveDiversity...................... 30 4.5SuboptimalDetectorforNonuniformD-STBC.......... 32 4.6ComparisonofDierentialGroupCodesandADierential AlamoutiCode.......................... 34 4.6.1DierentialAlamoutiCode................. 34 4.6.2AlamoutiCodewithDierentiallyEncodedSymbols... 36 5CONVOLUTIONALPRE-CODING................... 40 5.1IntroductiontoConvolutionalCodes............... 40 5.2ConvolutionalCodesAppliedtoSpace-TimeCodingSystem.. 42 5.2.1Hard-DecisionDecoding................... 43 5.2.2Soft-DecisionDecoding................... 43 6ANETWORKAPPLICATIONEXAMPLE............... 48 v

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7CONCLUDINGREMARKSANDFUTUREWORKS......... 55 REFERENCES.................................. 57 BIOGRAPHICALSKETCH........................... 60 vi

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LISTOFFIGURES Figure page 1{1Apoint-to-pointlink........................... 1 1{2Apoint-to-multipointlink........................ 2 3{1Anonuniform8-PSKconstellationobtainedfromastandard\uniform"QPSKconstellationbysplittingeachoriginalconstellation point\o"intoapairofnewpoints\ "............... 8 4{1Receivedsignallevelmayuctuatevastlyforasingleradiochannel. 10 4{2Totalchannelgainoftwoindependentfadingchannels........ 11 4{3Awirelesslinkwithmultitransmitandreceiveantennas....... 12 4{4OSTBCdecouplesaMIMOchannelinto n s numberofAWGN channels................................ 19 4{5Optimalvaluesof( ; )forthe R b =1, R a =1code,foragiven acceptableperformancedegradationofthebasicmessage.The resultsareobtainedviaminimizationoftheunionbound(4.36) andacorrespondingexpressionfortheerrorrateoftheadditionalmessage............................. 27 4{6Performancedegradationforthebasicmessageandperformance gainfortheadditionalmessageforthe R b =1, R a =1code. Theresultsareobtainedvia(4.36),alongwithacorresponding expressionfortheadditionalmessage.Thecurvesforthebasic messagearenormalizedrelativetothe\undisturbedcase"( = =0)andthecurvesfortheadditionalmessagearenormalized relativeto =0 : 2, =0 : 035.................... 28 4{7EmpiricalBERforthe R b =1, R a =1codewith =0 : 2, =0 : 035. 29 4{8EmpiricalBERforthe R b =2, R a =1nonuniformspace-time constellation.Theparameterswere =0 : 078, =0 : 018(found viaoptimizationof(4.36))...................... 31 vii

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4{9Monte-CarloSimulationoftheBERperformanceforthesuboptimalreceiver(4.49)and(4.50)inSection4.5,forthejointML receiverandforthebasicmessageonlysystemwith R b =2, R a =1code.Thecurves\ "and\ "overlap..... 38 4{10MonteCarloSimulationresultfortheBERperformanceof R b = 2, R a =1system,withdierentialgroupcodesanddierential Alamouticodes.Notethatthesolidanddashcurveswith\ marksoverlap............................. 39 5{1Asimplerate1 = 2convolutionalencoder................ 41 5{2Withrate-1/2convolutionalcoding,comparisonofBERperformanceareshownforhard-decisiondetection,soft-decisiondetectionwithconcentratedlikelihoodfunction,andsoft-decisionwith estimatednoisevariance........................ 47 6{17-cellfrequencyreusesystem...................... 49 6{2Typicalcelllayoutforawirelesstelecommunicationsystem..... 50 6{3Coverageareaforthebasic/additionalmessagesasafunctionof the\tolerable"performanceloss,forthe R b =1, R a =1code. Thehorizontalcurvesrepresenttherelativecoverageareaforthe R b =2codewithabasicmessageonly................ 53 6{4CoverageareafortheSISObasic/additionalmessagesasafunction ofthe\tolerable"performanceloss.ThehorizontalcurvesrepresenttherelativecoverageareafortheSISOrate-2codewitha basicmessageonly........................... 54 viii

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Figure1{1:Apoint-to-pointlink. 1

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Figure1{2:Apoint-to-multipointlink. system,whereasaradio/TVbroadcastisanexampleofapoint-to-multipointlink(thereforepoint-to-multipointlinksaresometimesreferredtoasbroadcastchannels).Point-to-multipointlinksarebecomingincreasinglyimportant.Forin-stance,theintroductionofDigitalAudioBroadcast(DAB)andHigh-DenitionTelevision(HDTV)haspioneeredawholeneweldofdigitalbroadcastingapplications.Asafurtherexample,itiswidelybelievedthatmuchofthenextgeneration'swirelessnetworkingwillbebasedonso-calledad-hocnetworks,whereitmaybenecessaryformultipleunitstolistentoonemessageatthesametime.Also,inconventionalcellcommunicationsystemusingdirectionaloradaptiveantennas,itissometimesnecessarytobroadcastamessagetotheentirecell.Therearetwomajordierencesbetweenpoint-to-pointandpoint-to-multipointcommunicationlinks.First,apoint-to-pointconnectioncanbeoptimizedforagiventransmitter-receiverpair.Forinstance,acellularsystemusuallyemployspowercontroltechniquesthatadjustthetransmittedpowertominimizethepowerconsumption,reducetheamountofco-channelinterfer-ence,andatthesametimeensurethatthereceivedsignalstrengthexceedsa

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certainthreshold.Second,incontrasttoapoint-to-pointcommunicationlink,forabroadcasttransmissionallreceivershavedierentcapabilitiestodecodethetransmittedmessage.Thisissobecausethedierentreceiversexperienceingen-eralverydierentradiolinkqualities(forinstance,duetovaryingpropagationconditions).Moreover,sincequalitycanusuallybetradedforcost,thereceiversthemselvesmayhavedierentinherentabilitiestodecodethetransmittedmessage.Thefactthatdistinctreceivershavedierentcapabilitiesofdecodingames-sagesuggeststhatthetransmittedsignalshouldconsistofseveralcomponentswhichareofdierentimportanceforthereconstructionofthemessage(andthereforehaveaninherentlydierentvulnerabilitytotransmissionerrors).Thisideahasleadtotheconceptoflayeredsourcecoding(e.g.,[1,2,3,4]),whichisnowamaturetechniqueemployedinmanymultimediastandards.Forinstance,theimagecodingstandardJPEG-2000andthevideocodingstandardMPEG-4usewhatissometimesreferredtoas\negranularityscalability,"whichenablesagradualtradeobetweentheerror-freedatathroughputandthequalityofthereconstructedimageorvideosequence[5].SuchprogressivesourcecodingmethodsarealreadyinuseinmanyInternetapplicationswheredataratecanbetradedforquality,andtheyareexpectedtoplayaninstrumentalroleforthenextgenerationofwirelessstandardstoprovideubiquitousaccessbothtotheInternet,andtodiversesourcesofstreamingvideoandaudio.Thetopicofthisthesisisspace-timecodingforbroadcastchannelswhenlayeredsourcecodingisused.PreviousworksonthetopicincludeMemarzadehetal.[6],whichdiscussed\beamforming"techniques(hencerequiringfeedbackofchannelstateinformation)foramultiusersystem.InMemarzadehetal.[6],andtwokindsofbeamformingtechniquesarediscussed.Zero-forcingbeamformingis

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introducedwhichorthogonalcodewordsareusedforeachusersothatinter-userinterferencecanbefullyeliminatedjustasinanidealCDMAsystem,providedthatexactchannelstateinformationisavailableatthetransmitter.Anotherwayisthesingleuseroptimalbeamformer,forwhichnoattemptismadetoavoidinter-userinterference.Thisissuitableforthescenariowhenthechannelstateinformationatthetransmitterisnotveryaccurate,asinterferencesuppressionisnoteective.Kuoetal.[7]proposedalayeredspace-timecodingschemeassumingdierentreceiversmayhavedierentnumbersofreceiveantennas,andthatareceiverwithmorereceiveantennascandecodemorelayersofmessages.Inthisthesis,wesuggestanewlayeredspace-timecodingschemethatdoesnotrequireanychannelknowledgeatthetransmitter,andwhichisconstructedstartingfromacriterionthatattemptstominimizetheerrorrate.Ournewcodesalsosatisfytwootherimportantdesigngoals.First,theycanbeencoded(anddetected)dierentially.Second,theyareentirelybasedonphase-shiftkeying(PSK)modulationandconsequentlythetransmittedsignalhasconstantenvelopeatalltimes.Thecodesproposedinthispapercanthereforebeseenasamultidimensionalextensionofatransmissiontechniqueforsingle-antennasystemsinPursleyetal.[8].

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Withlayeredsourcecoding(forinstanceasmentionedabove,JPEG-2000andMPEG-4),morecapablereceivers,forexamplewithhighersignal-to-noise-ratio(SNR),canachieveahigherdataratebydecodingthemessagescontainedinalllayerswhilethelesscapablereceiverscanonlydecodethemessageinthebottomlayer.Thusthe\mostimportant"informationshouldbeconveyedviathebottomlayerwhile\lessimportant"informationcanbetransmittedviahigherlayer(s).Anexampleofamodulationschemedesignedforlayeredsourcecodingisnonuniformphase-shift-keyingmodulation(PSK),whichwasrstintroducedbyPursleyandShea[8](seealsoPursleyetal.[12,13]).Theideaisintuitivelyappealing.StartingwithastandarduniformPSKconstellation,whichcontainsdataforthe\basic"layer,anonuniformPSKcodeconstellationisobtainedbyaddingasmalladditionalphaseshifttoeachoriginalconstellationpoint,whichcontainstheinformationforthe\additional"layer.ThisisillustratedinFigure3{1,whenthebasicmessageisconveyedviaaQPSKconstellation.AcapablereceiverwithhighSNRcandistinguishamongalleightconstellationpointswhichmeansthatboththebasicandtheadditionallayermessagescanbedecoded,whereastoalesscapablereceivertheconstellationmayappearlikeablurredQPSKconstellationandhenceitmayonlybeabletodistinguishbetweenthedierent\large"phaseshifts;thusonlythebasicmessagecanbedetectedaccurately.Bysuchaconstructiontheerrorratesassociatedwiththebasicandtheadditionalmessagearedierent.Consequently,theadditionalmessagecancarryinformationthatisoflessimportanceforthereconstructionofthetransmittedmessagethanthebasicmessageis.Theerrorprobabilityoftheadditionalmessagecanbeeasilyadjustedbychoosingdierentvaluesoftheadditionalphaseshift.Obviously,alargerresultsinabetterperformance

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Figure3{1:Anonuniform8-PSKconstellationobtainedfromastandard\uni-form"QPSKconstellationbysplittingeachoriginalconstellationpoint\o"intoapairofnewpoints\". fortheadditionalmessageatthecostofadeteriorationoftheperformanceforthebasicmessage.ThemainvirtueofthisnonuniformPSKencodingschemeisthatthesignalingenvelopeisconstantwhilemulti-layeredmessagescanbetransmittedwithdierentandadjustableerrorperformance.

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4.1 Error Performance of STBC SystemsAconventionalwirelesssystemwithasingletransmitandreceiveantennasuersfromalargeperformancedegradationduetothefadingoftheradiochannel.Fadingistheuctuationinreceivedsignallevel.Itismainlyduetochangesinthepropagationenvironment,suchasweatherandmovingreectiveobjects/obstructions.Fadingcanbeclassiedasfastfadingorslowfading,dependingonhowrapidlythechannelimpulseresponsechanges.Forafastfadingchannel,thechannelimpulseresponsechangesrapidlywithinonesymbolduration,whichmeansthatthecoherencetimeofthechannelTc(whichisameasureofthetimedurationthatthechannelimpulseresponseisquasi-static)issmallerthanthesymbolperiodofthetransmittedsignalTs>Tc(4.1)Foraslowfadingchannel,thechannelimpulseresponsechangesveryslowlycomparedwithonesymbolduration,whichmeansthatthechannelcanbeseenasquasi-staticformanysymboldurations,thatisTsTc(4.2)Figure4{1showsanexampleofthefastfadingofthegainofasingleradiochannel.Thetwocurvesrepresenttwochannelsencounterindependentfastfading.Wecanseethatthereisahighriskthatasinglechannelmayencounter 9

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Figure4{1:Receivedsignallevelmayuctuatevastlyforasingleradiochannel. adeepfadeatacertaintime.Duringdeepfadethetransmissionqualitymaybeverypoororthetransmissionmaysimplybeimpossible.Tocounteractthefadingradioenvironmentamultipletransmitandreceiveantennasystem(orMIMO)isoneattractivepossibility.Theadvantageofamultipletransmitandreceiveantennaisnotdiculttocomprehendintuitively:Therearemorethanonechannelwhicharenotfullydependent,sothepossi-bilitythatallofthemencounteradeepfadingatthesametimeissmall;thusthedeepfadingproblemmaybesolved.Figure4{2showsthetotalchannelgain(denotetheindividualchannelgainsbeh1andh2,totalchannelgainisthenequaltojh1j2+jh2j2)oftwoindependentfadingchannels.Itcanbeseenthattheriskofthetotalchannelgainencounteringadeepfadeislargelyreduced.

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Figure4{2:Totalchannelgainoftwoindependentfadingchannels. Thecorrespondingencodingschemeforsuchamultitransmitandreceiveantennasystemiscalledspace-timecoding.Thisisanemergingtopicthatnowattractsmanyresearchers'attentionsinceitsintroductionaboutadecadeago.Themainvirtueofsuchamultitransmitandreceiveantennasystemisthathigherdiversityordercanbeachieved.ThediversityordercanbedenedimplicitlybyBit-Error-Rate/(SNR)diversity(4.3)whichmeansthatthebit-error-ratedecreasesasfastastheSignal-to-Noise-Ratio(SNR)tothepowerofminusthediversityorder.ForahighSNRtheadvantageofahighdiversityordercanbesubstantial.ConsideraMIMOsystemwithnttransmitantennas,nrreceiveantennas(seeFigure4{3),andassumeforsimplicitythatthepropagationchannelis

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Figure4{3:Awirelesslinkwithmultitransmitandreceiveantennas. linear,time-invariantandfrequencyat.LetHbeannrntmatrixwhose(m;n)thelementcontainsthechannelgainbetweentransmitantennanandreceiveantennam,andsupposethatacodematrixXofdimensionntNistakenfromamatrixconstellationXandtransmittedbysendingitsNcolumnsviathentantennasduringNtimeepochs.IfthesignalreceivedduringthesameNtimeintervalsisarrangedinannrNmatrixY,thenY=HX+E(4.4)whereEisannrNmatrixofnoise.Throughoutthisthesis,weshallmakethesomewhatstandardassumptionthattheelementsofHandEareindependentandzero-meancomplexGaussianwithvariances2and2,respectively;hencethechannelisRayleighfading.TheShannonchannelcapacityofsuchaMIMOsystemisawell-knownresult,anditcanbeshownthat(fordetailssee[14]chapter3andthereferences

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therein)C(H)=Blog2jI+1

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dierentcodematrixX6=X0occursisP(errorjH)=PkYHXk2kH~Xk2HexpkH~Xk2 42kH~Xk21

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TreatingHasadeterministicunknown,wecanminimizethislikelihoodfunctionwithrespecttoH.NoticethatkYHXk2=YHXHHH2=(XH+?XH)(YHXHHH)2=XH(YHXHHH)2+?XH(YHXHHH)2+2ReTr(YHX)XH?XH(YHXHHH)| {z }0=XHYHXHHH2+?XHYH2(4.14)HereXH=XH(XXH)1XistheorthogonalprojectorofXH.Thefunctionin(4.14)isminimizedwhenXHYHXHHH=0()XH(XXH)1XYHXHHH=0()XH(XXH)1XYHHH=0(4.15)IfXhasfullrankthentheaboveleadstotheresult(see,e.g.,[15]and[14]appendixB)H=YXH(XXH)1(4.16)(ItmaybearguedthatifweknowsomesortofstatisticalpropertiesofH,weshouldtreatHasastochasticvariablewhenitentersthelikelihoodfunction.Thisisintuitivesinceweshouldincorporateallknowninformationintothedetectortogetthebestresult.ButinLarssonetal.[16](seealsoLarssonetal.[14]exercise9.7)itisshownthatbothapproachesindeedgivethesameresult.Inthispaperwewillsticktothedeterministicchannelapproachforsimplicity.)

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Inserting(4.16)into(4.13)yieldsminX2XkYYXH(XXH)1Xk2()minX2XkY?XHk2()maxX2XkXHYHk2(4.17)Let=XH0XH(4.18)NotethatthematrixisHermitian,butnotnecessarilypositivedenite.AlsonotethatXH0=(XH0XH)XH0=XH0XHXH0=?XHXH0(4.19)Hence,since?XHXH0hasfullrankbyassumption,thematrixX0?XHXH0ispositivedenite.TheprobabilitythattheMLdetectormakesamistakeisP(errorjH)=PkXHYHk2>kXH0YHk2H=PTr(HX0+E)(HX0+E)H<0H=P2ReTrHX0EHg+TrfEEHg
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ThetermEEHisofhigherorderandmaybeneglectedinanasymptoticanalysis.Inparticular,foragivenH,thevarianceofitselementsisoftheorder4,whereastheelementsofHX0?XHEHhaveavarianceoftheorder2.SincedQ(x) 21 42k?XHXH0HHk21

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Aspecialsubclassofspace-timecodesisthelinearorthogonalspace-timeblockcodes(OSTBC).ThecasewhenOSTBCappliedtoa2-TXsystemturnsouttobethewell-knownAlamouticode[17].ThemainvirtueofOSTBCisthattheyachieveadiversityofordernrntforasystemwithnttransmitandnrreceiveantennas,atanalmostnegligiblecom-putationalcost.Specically,forsuchasystem,asetofnssymbolsfs1;:::;snsgareencodedintoanntNmatrixthathasthefollowingstructureXOSTBC=nsXn=1snAn+snBn(4.26)ThecolumnsofXOSTBCaretransmittedviathenttransmitantennasduringNtimeintervals.In(4.26),fAn;BngarematriceschosensuchthatXOSTBCXHOSTBC=nsXn=1jsnj2I(4.27)forallfsng.Ingeneral,theroleoftheorthogonalitycondition(4.27)andtheassociateddesignoffAn;Bngisrelativelywellunderstoodbynow;forinstance,(4.27)impliesthatthesymbolsfsngcanbedetectedindependentlyofeachother[14],sowiththeuseofOSTBC,theMIMOchannelcanbedecoupledintonsnumberofsingleadditive-white-gaussian-noise(AWGN)channels(SeeFigure4{4). 4.2 Dierential Modulation for MIMO SystemsSpace-timemodulationmatricesthatcanbeencodeddierentiallyaresometimesofspecialinterestsincesuchcodescaneasilybedemodulatednonco-herently.Dierentialcodeswithuniformerrorprobabilitieshavebeenstudiedbymanyauthors(see,e.g.,[18,19,20,21,22]forsomeprominentexamples)andsomeofthemcanbeseenasanextensionofdierentialPSKtoMIMOsystems.Ingeneral,iftisthetimeindexandiffU(t)gisasequenceof(square)

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Figure4{4:OSTBCdecouplesaMIMOchannelintonsnumberofAWGNchan-nels.

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information-bearingmatrices,thendierentialencodingobtainsthetransmittedcodematrixX(t)attimetviaX(t)=X(t1)U(t)(4.28)whereX(t1)isthematrixtransmittedattimet1.Suchanencodingusuallyisonlymeaningfulundercertaincircumstances,forinstanceifU(t)isunitary(inwhichcaseX(t)becomesunitaryforalltaswell,givenaunitary\initialmatrix"X(0)).Byconsideringtwo(ormore)receivedmatricesY(t)andY(t1)simultaneously,noncoherentdetectionispossible.Forexample,ifweconcatenatethematricesreceivedattimet1andt,andassumethatthechannelHremainsconstantoverthesetwotimeintervals(towithinpracticalaccuracy),wecanwrite(using(4.4)and(4.28))Y(t1)Y(t)=HX(t1)HX(t)+E(t1)E(t)=HX(t1)HX(t1)U(t)+E(t1)E(t)=HX(t1)IU(t)+E(t1)E(t)(4.29)whereHX(t1)canbeseenasanunknown\eective"channelmatrixandIU(t)isaneectivecodematrix.ProvidedthatIU(t)issuchthatthecorrespondingdeterminantin(4.25)isnonzero,noncoherentdemodulationispossible.Byapplyingthedecisionrule(4.17)tothedatamodel(4.29),the

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detectionrulefordierentialdemodulationbecomesminU(t)2XY(t1)Y(t)?[IU(t)]H2()minU(t)2XTr8><>:Y(t1)Y(t)?[IU(t)]H264YH(t1)YH(t)3759>=>;()minU(t)2XTr8><>:Y(t1)Y(t)264IU(t)UH(t)I375264YH(t1)YH(t)3759>=>;()maxU(t)2XReTrU(t)YH(t)Y(t1)(4.30)whereTrfgstandsforthetraceofamatrixandwhereweusedthefactthattheinformation-bearingmatrixU(t)isunitary.Theassociatedaverageerrorprobability,i.e.,theprobabilitythatU(0)isincorrectlydetectedasUforthisdierentialdetectionschemecanbeboundedbyusing(4.25)EhP(U(0)!U)iIU(0)?[IU]H264IU(0)H375nr2 2(UU(0)H+U(0)UH)nr2 4.3 Nonuniform Space-Time CodesInspiredbythenonuniformPSKcodesdiscussedinSection3,ournewnonuniformspace-timecodesarebasedondierentialencodingoftheproductofaunitarycodematrixUb2Xbassociatedwithabasicmessage,andanotherunitarycodematrixUa2Xacorrespondingtoanadditionalmessage.Thusthetransmittedcodematrixattimetisgivenby(cf.(4.28))X(t)=X(t1)Ub(t)Ua(t)(4.32)

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Using(4.31)alongwiththefactthatUbUaisunitary,wendthattheaverageprobabilitythatatransmittedmessagepair(U0b,U0a)ismistakenforanotherpair(Ub,Ua)canbeboundedbyEP(U0b;U0a!Ub;Ua)I1 2UbUaU0HaU0Hb+U0bU0aUHaUHbnr2 4.3.1 Design CriteriaSincetheadditionalmessagecanbedecodedonlyathighSNR,thematricesfUagassociatedwiththeadditionalmessageshouldapproximatelybeclosetotheidentitymatrix.Inspiredbythisobservation,wecanrstconsiderthedesignoffUbg,treatingthepresenceofUaasanunmodellednoise-likedisturbanceterm.Doingso,wecanapproximatelyboundtheerrorprobabilityforUbalone(assumingdierentialdetection)byEP(U0b!Ub).I1 2U(0)bUHb+UbU(0)Hbnr2

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shouldbeareasonableassumption,sinceifitisnottruethendecodingofUaisprobablyoflessinterestanyway.AssumingthatUbisknown,thedemodulationofUaisessentiallyanothernoncoherentdetectionproblem.ToobtainacriteriaforthedesignoftheconstellationfUag,wewanttoformanerrorboundonUaandaverageitoverallpossiblebasicmessagesfUbg.Usingthebound(4.25)fornoncoherentdetectionappearstoyieldcriteriathatareverydiculttouse.Asasuboptimalapproachweusedinsteadthebound(4.24)forcoherentdetection.DoingsoresultsinthefollowingcriterionEP(U0a!Ua)1 4.3.2 Union Bound on the Error ProbabilityBytheunionbound,theprobabilityforthebasicmessagetobeinerror(averagedoverallpossiblepairsofadditionalmessages)canbeboundedbyE[P(basicmessageinerror)]1 2U(k)bU(r)aU(s)HaU(n)Hb+U(n)bU(s)aU(r)HaU(k)Hbnr2

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Likewise,theerrorratefortheadditionalmessage(averagedoverallpossiblepairsofbasicmessages),canbeboundedbyasimilarexpressionE[P(additionalmessageinerror)]1 2U(k)bU(r)aU(s)HaU(n)Hb+U(n)bU(s)aU(r)HaU(k)Hbnr2 4.4 Design Examples 4.4.1 Code for 2 TX AntennasWerstconstructacodewheretherateforthebasicmessageisRb=1bit/sec/HzandtheratefortheadditionalmessageisRa=1bit/sec/Hzaswell.WetakethebasicmessageUbtakenfromthefollowingsetXb=8><>:2641001375;2640110375;2640110375;26410013759>=>;(4.38)Theconstellationin(4.38),whichisuniformandpossessescertainoptimalityproperties,isduetoHughesetal.[18]andwascalled\BPSK"therein.BasedonthedesignrulesinSection4.3.1,wehavehandcraftedthefollowingconstellationofmatricesfortheadditionalmessageUaXa=(264ei00ei375;264ei00ei375;264ei00ei375;264ei00ei375)(4.39)where(;)aredesignparameters(tobediscussedbelow).

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Theconstellationmayappeartobesomewhatarbitrarybutitpossessessomenicefeatures.First,allmatricesin(4.39)areunitary,thisisarequirementforMIMOdierentialencoding.Second,itissymmetricanduniformwhichmeansthattheerrorprobabilityperformanceshouldbethesameforallconstel-lationpoints.AlsosimilarlytothenonuniformPSKencodingforasinglechannel(cf.Section??),multiplicationwiththematricesin(4.39)canbeinterpretedasaddingasmallphaseshifttotheelementsofthebasicmessageUb.Wemayalsothinkofotherformsofmatricesastheadditionalmessage.Forinstance,considerthefollowinganti-diagonalmatrixUa=2640eiei0375(4.40)Butweimmediatelyndthatthisdoesnotwork.Consider,forexample,Ub=2641001375(4.41)Then,UbUa=26410013752640eiei0375=2640eiei0375(4.42)ItisseenthatmultiplyingthisadditionalmessageUanotonlyintroducesasmallphaseshiftbutalsointerchangesthecolumnsoftheoriginalbasicmessageUb.Thisisnotacceptablesinceourdesigngoalistohavetheadditionalmessageonlyposeasmalldisturbancetothebasicmessagesothatforthoseless-capablereceiverscanstilldetectthebasicmessageevenwithoutknowledgeoftheexistenceoftheadditionalmessage.TheUaheresimplydestroystheoriginalbasicmessageconstellationsoitisnotfeasible.

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Letusnowreturntoourdesigntask.Ifwetaketheinitialtransmitmatrix,somewhatarbitrarilyX(0)=1

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Figure4{5:Optimalvaluesof(;)fortheRb=1,Ra=1code,foragivenac-ceptableperformancedegradationofthebasicmessage.Theresultsareobtainedviaminimizationoftheunionbound(4.36)andacorrespondingexpressionfortheerrorrateoftheadditionalmessage. furtherillustratedinFigure4{6,whereweshowhowtheperformanceassociatedwiththebasicandadditionalmessagesvarieswith(;).Figure4{7showstheempiricalbit-error-rate(BER),obtainedviaMonte-Carlosimulation,forthecodedescribedaboveusing=0:2,=0:035andMLdecoding.Thesolidlines(\|")showtheperformanceofdierentialnonuniformBPSKforaconventionalsystemwithnt=1transmitantennaandasinglereceiveantenna(thisisessentiallyaspecialcaseofPursleyetal.[8]).Thedashedlines(\--")showtheperformanceforasystemwithnt=2(andasinglereceiveantenna)usingthenewcodepresentedabove.Forthecurveswithoutmarks,onlyabasicmessageistransmitted.Thecurveswithmarksshowtheperformancewhenbothabasicandanadditionalmessagearetransmitted:the

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Figure4{6:PerformancedegradationforthebasicmessageandperformancegainfortheadditionalmessagefortheRb=1,Ra=1code.Theresultsareobtainedvia(4.36),alongwithacorrespondingexpressionfortheadditionalmessage.Thecurvesforthebasicmessagearenormalizedrelativetothe\undisturbedcase"(==0)andthecurvesfortheadditionalmessagearenormalizedrelativeto=0:2,=0:035. curvesmarkedwith\"showtheBERforthebasicmessage,andthecurveswith\"showtheBERfortheadditionalmessage.Clearly,thetransmitdiversitysystemoutperformstheconventionalone{observe,inparticular,thedierentslopesoftheBERcurvesbothforthebasicandfortheadditionalmessage.Thesimulationalsoconrmsthatthetransmissionofanadditionalmessageincursasmallperformancedegradationforthebasicmessage.

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Figure4{7:EmpiricalBERfortheRb=1,Ra=1codewith=0:2,=0:035.

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4.4.2 CodeToobtainacodewithahigherinformationrateforthebasicmessage,wenexttakeUbfromthefollowingalgebraicgroupof16matrices(whichisduetoHughesetal.[18]aswell)generatedbyXb=*264ej=400ej=4375;2640110375+(4.46)Thenotationin(4.46)meansthatallpossiblematricescanbegeneratedbychoosinganyarbitraryintegersMandNinthefollowingexpression264ej=400ej=4375M2640110375N(4.47)Theconstellationusedfortheadditionalmessageischosentobethesameas(4.39),sowenowhaveasystemwithrateRb=2forthebasicmessageandrateRa=1fortheadditionalmessage.ThecorrespondingsimulatedBERisshowninFigure4{8. 4.4.3 Receive DiversityTodemonstratethetransmitdiversityachievedforthe2-TXsysteminthesimplestway,ourresultsabovearebasedonthe1-RXcase.Althoughofthis,ournewlyproposednonuniformSTBCcodesarenotonlyusefulfor1-RXcasebutalsoforageneralnr-RXcase.Thisisobvioussinceourtheoreticalderivationsabovearenottailoredforthe1-RXcase,henceitshouldworkautomaticallyforageneralnr-RXcase.Alsowecouldexpectadiversityorderofnrntcanbeachieved.Thisissosincewecanseethedatamodelofanr-RXnt-TXsystemas

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Figure4{8:EmpiricalBERfortheRb=2,Ra=1nonuniformspace-timecon-stellation.Theparameterswere=0:078,=0:018(foundviaoptimizationof(4.36)).

PAGE 41

whereYkfork=1;;nristhe1Nreceiveddataatthekthreceiveantenna,Hkisthe1ntchannelmatrixassociatedwiththekthreceiveantenna,Ekisthe1ntnoisematrixatthekthreceiveantenna.Asusual,forsimplicityifweassumethechannelmatricesassociatedwithnrdierentreceiveantennasHk(fork=1;;nr)areindependentandthenoiseiswhite(E1;;Enrareuncorrelated),thenthenr-RXnt-TXcanbebrokendownintonrnumberofindependent1-RXnt-TXsystem.EachreceiveantennahassucientdatatodetectthetransmittedmatrixXsotheycanworkindividuallyandachievetransmitdiversityofordernt.TofurtherutilizereceivediversitywecanemployjointMLdetectionacrossallnrindependentreceiveantennasthusincreasingtheoveralldiversitybyafactorofnr.Sincethenrreceiveantennasareindependentatthereceiversite,sooptimizingthespace-timeencodingschemeforasinglereceiveantennaisequivalenttooptimizingasystemwithnrindependentreceiveantennas. 4.5 Suboptimal Detector for Nonuniform D-STBCTodetectboththebasicandtheadditionalmessagesinajointMLfashionwehavetosearchthroughalljXbjjXajpossiblecombinationstondthecombinationofUbandUawhichmaximizethecostfunctionin(4.30).Thismaybecomputationallyburdensomeiftheconstellationsizesofbothlayersarelarge.ApossibleremedytothisproblemistouseasuboptimaldetectionruledescribedasfollowsStep1.Detectthebasicmessageonlyusing(4.17),neglectingthepresenceoftheadditionalmessage^Ub(t)=minUb(t)2XbY(t1)Y(t)?[IUb(t)]H2()maxUb(t)2XbReTrUb(t)YH(t)Y(t1)(4.49)

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tothejointMLdetectionbecauseifonlythebasicmessageisneeded,thecomputationalloadwillbefurtherreducedtojXbj=16(forthisexample).Weexpectthatthereissomeperformancelossassociatedwiththissubopti-maldetectionschemesinceitdoesnotexploitknowledgeofthestructureoftheadditionalmessageintheStep1.TheMonte-CarlosimulationresultinFigure4{9showstheperformanceofthesuboptimaldetectorfortheRb=2,Ra=1codecomparedtotheperformanceofthejointMLdetector.Thedetectionofthebasicmessagegivesasmallperformanceloss(0:3dB)comparedtothejointMLdetector,whereasthedetectionoftheadditionalmessageis(towithinpracticalaccuracy)equaltothatofthejointMLdetector. 4.6 Comparison of Dierential Group Codes and A Dierential Alamouti Code 4.6.1 Dierential Alamouti CodeThemainadvantageofthenonuniformspace-timegroupcodesintroducedinSection4.4isthatthesignalenvelopeisconstant.However,alsoforthisreasontheycannotachievethemaximumpossibleperformance(intermsofBER)becausetheydonotmakeuseofanyamplitudeinformationinthesignal.Onthecontrary,orthogonalspace-timeblockcodes(OSTBC)shouldprovidebetterperformance(BER)sincetheyarealreadyoptimizedbyconstruction[17,23,14].For2TXantennasOSTBCreducestoAlamouticode.TheconstructionofanAlamouticodematrixisasfollowsU(t)=1

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performancesofthedetectionoftheadditionalmessageareequalbothwhentheAlamoutiandthegroupcodes(4.39)areusedasthebasicmessagecarrier. 4.6.2 Alamouti Code with Dierentially Encoded SymbolsItmaybearguedthatboththeconstantsignalingenvelopeandmaximumBERperformancecanbeachievedatthesametimebyusingsymbol-wisedierentialencodingandformingtheAlamoutimatrixafterwards.However,unfortunatelythisencodingschemedoesnotwork.Thisisdemonstratedasfollows.Supposethatweemploysymbol-wisedierentialencodingx1(t)=x1(t1)s1(t)x2(t)=x2(t1)s2(t)(4.54)andusetheseelementstoformtheAlamoutimatrixX(t1)=264x1(t1)x2(t1)x2(t1)x1(t1)375X(t)=264x1(t)x2(t)x2(t)x1(t)375=264x1(t1)s1(t)x2(t1)s2(t)x2(t1)s2(t)x1(t1)s1(t)375(4.55)Theabovecodematrixhasthepropertythatjx1(t)j=jx2(t)j=js1(t)j=js2(t)j=1foralltandhencethesignalenvelopeisconstantforbothtransmitantennasallthetime.AlsoX(t)isunitaryforalltsinceitisanAlamoutimatrix.Theobservationmodel(4.29)becomes(cf.Section4.2)Y(t1)Y(t)=HX(t1)X(t)+E(t1)E(t)=HX(t1)IXH(t1)X(t)+E(t1)E(t)(4.56)

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Employingthenoncoherentdetectionrule(4.30totheabovemodel(4.56)thefollowingdetectionruleisobtainedmaxs1(t);s2(t)2SRenTrn(XH(t1)X(t)YH(t)Y(t1))oo()maxs1(t);s2(t)2SRe8<:Tr8<:24x1(t1)x2(t1)x2(t1)x1(t1)3524x1(t1)s1(t)x2(t1)s2(t)x2(t1)s2(t)x1(t1)s1(t)35YH(t)Y(t1)9=;9=;()maxs1(t);s2(t)2SRe8<:Tr8<:24s1(t)+s2(t)x1(t1)x2(t1)(s2(t)s1(t))x1(t1)x2(t1)(s1(t)s2(t))s1(t)+s2(t)35YH(t)Y(t1)9=;9=;(4.57)Theabovemaximizationproblem(4.57)cannotbesolvedunlessx1(t1)andx2(t1)areknown.Forexample,considerthefollowingtwosetsofvaluesfs1(t);s2(t);x1(t1)x2(t1)g=f1;1;1g(4.58)andfs1(t);s2(t);x1(t1)x2(t1)g=f1;1;1g(4.59)Thesemakethemetricin(4.57)equaltoRe8><>:Tr8><>:2640220375YH(t)Y(t1)9>=>;9>=>;(4.60)whichimpliesthatthesetsfs1;s2g=f1;1gandfs1;s2g=f1;1garenotidentiableifx1(t1)andx2(t1)areunknown.

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Figure4{9:Monte-CarloSimulationoftheBERperformanceforthesuboptimalreceiver(4.49)and(4.50)inSection4.5,forthejointMLreceiverandforthebasicmessageonlysystemwithRb=2,Ra=1code.Thecurves\"and\"overlap.

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Figure4{10:MonteCarloSimulationresultfortheBERperformanceofRb=2,Ra=1system,withdierentialgroupcodesanddierentialAlamouticodes.Notethatthesolidanddashcurveswith\"marksoverlap.

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5.1 Introduction to Convolutional CodesInPursleyetal.[12],convolutionalcodingareintroducedforextraerrorprotectionbeforethedatasymbolsarefurtherencodedbythenonuniformPSKmodulation.ForcompletenessweintroduceherethebasicconceptofconvolutionalcodingasatooltoimprovetheBERperformance.Convolutionalcodingisakindofforwarderrorcorrection(FEC)technique,whichmeansthatnofeedbackchannelisrequiredincontrastto,e.g.AutomaticRepeatRequest(ARQ).Theideaofconvolutionalcodingissimple:Toimprovetheerrorperfor-manceofatransmissionlinkbyaddingsomecarefullydesignedredundantbitstothedatabeforeitistransmittedthroughthechannel.Thisprocessofaddingredundancyisknownaschannelcoding.Incontrasttoblockcodeswhichoperateonlargemessageblocks,convolutionalcodesoperateonacontinuousstreamofdatabits.ConvolutionalcodingisaFECtechniquewhichisespeciallysuitabletochannelswhicharecontaminatedbyadditivewhitegaussiannoise(AWGN).Byusingconvolutionalcoding,BERperformancecanbeimprovedsignicantlyatthesameSNR,atthecostofloweringthedatarate.Meanwhilethisratelosscanbecompensatedforbyincreasingtheconstellationsize.Therearetwoimportantparametersforaparticularconvolutionalcodes,namelythecoderateandtheconstraintlength.Thecoderate,k=n,issimplytheratioofthenumberofdatabitsinputtotheconvolutionalencoder(k)tothe 40

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Figure5{1:Asimplerate1=2convolutionalencoder. numberofoutputbitsbytheconvolutionalencoder(n),canalsobeameasureoftheeciencyofthecode.Theconstraintlength,K,denotesthe"length"or"memory"oftheconvolutionalencoder,thatis,eachoutputbitisaectedbyKpreviousinputbits.Foryears,convolutionalcodinghasbeenthepredominantFECtechnique.Butthereisatradeoofemployingconvolutionalcoding.Forexample,usinga1=2rateconvolutionalcodes,thedataratewillbedeductedtohalf,providedthatthemodulationtechniqueisthesame.Thisissimplybecausewithrate1=2convolutionalencoding,twochannelsymbolsareneededtobetransmittedperonedatabit.Buttheadvantageisthat,theconvolutionalcodingcanimprovetheerrorperformanceatthesameSNR,atthecostthatthedataratewilldecreasebyafactorofn=k.Figure5{1showsatypicalrate1=2convolutionalencoder,builtwithtwoshiftregistersandtwomodulo-2adders.

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Oneshouldpayattentionthataninterleaverisusuallyusedtointerleavetheencodeddatabitstreamfromtheoutputoftheconvolutionalencoder.Thebenetofinterleavingthedatabitsisthatitcanprovidetimediversity:consecutivebitsarelikelytobecontaminatedbyatemporarystronginterferenceornoise,orsustainedbydeepfadingchannelenvironment,thiscanmaketheerrorcorrectingalgorithmattheconvolutionaldecoderfailstocorrectbiterrorsandrecoverthecorrectinformation.Withsucientinterleaving,theoriginallyconsecutivebitsarenowseparatedfarapartastheyarebeingtransmittedthroughthewirelesschannel,sothatwecanreasonablyassumethatthebitsnearbyeachotherafterde-interleavinghaveundergoneindependentfadingandareaectedbyindependentnoiseandinterference,thisisdesirabletotheconvolutionaldecoderasconsecutivebitsarenotlikelytoencountererrorssimultaneously.AlsothedesigntaskoftheMLdetectoratthereceiverisalsosimplied. 5.2 Convolutional Codes Applied to Space-Time Coding SystemWearegoingtoinvestigatetheerrorperformanceofthesystemifacon-volutionalencoderisemployedtotheinformationbitstreamspriortoSTBCmapping.ItisexpectedthattheBERperformancecanbeimprovedsubstan-tiallybysuchkindofpre-codinginanadversechannelenvironment.Thissimplytradestheerrorperformancewithdatarate.Tostartwith,wechooseasimplerate-1=2convolutionalencoderwithconstraintlengthK=3,generatorsare5;7inoctal.AsforourRb=1;Ra=1code,therearetwobitscarriedinthebasiclayerandtwobitscarriedintheadditionallayerforeachtransmittimeinterval,lets1,s2denotethetwobitscarriedinthebasiclayerands3,s4denotethetwosbitscarriedintheadditionallayer,respectively.Andnowwehavefourseparate

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(independent)bitstreams,forexampletheymaylooklikethefollowings1=0;1;1;0;1;1;0;1;1;s2=1;1;0;1;1;0;0;0;1;s3=0;0;0;1;0;1;1;1;1;s4=0;0;1;0;0;1;0;1;0;(5.1)Wemodelthes1,s2,s3,s4asindependentbitstreamswithrandom0and1.Foreachbitstreamwepassitthroughastand-alonebinaryconvolutionalencoderstatedabove.Afterthat,wewouldhavefourencodedbitstreams,denotedbys01,s02,s03,s04.Noticethattheencodedstreamsarenotofthelengthoftherawdatastreams,fortherate-1=2encoderusedinourstudyhere,thelengthoftheencodedstreamsisdoubled,wesacricethedataratebyhalftotradeforbetterBERperformance.ThefourencodedbitstreamsarethenbeingpassedthroughtheSTBCmapperasbeforeandsentviatheMIMOsystem. 5.2.1 Hard-Decision DecodingWegettheestimatedbitstreams^s10,^s20,^s30,^s40attheoutputofthehard-decisionsymboldetector,wethenpassthemthroughtheViterbidecodertodecodethembackintotherawdatastreams.BERperformanceofsuchahard-decisiondecodingsystemisshowncomparingtotheun-encodedsystem. 5.2.2 Soft-Decision DecodingForasoft-decisiondecoder,theinputisnolongeraprecise0or1binaryestimationbutthelikelihoodratioP(c=1)=P(c=0)ofeachbits.Tocalculatethelikelihoodratioofeachindividualbitwehavetoconsiderthereceiverstructure.Forthedierentialencodinganddecodingsysteminsection4.2,todecodethedierentiallyencodedinformationmessage,thereceiver

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hastoconsidertwoconsecutivereceiveddatablocksY(t1)Y(t)| {z }Ye=HX(t1)| {z }eectivechannelHeIU(t)| {z }Ue+E(t1)E(t)| {z }Ee(5.2)whereYeandUestandforthe\eective"receiveddatablockandthetrans-mittedmatrix,respectively.TodetectthetransmittedmatrixUegivenYe,thenoncoherentdetectiontechniquediscussedinsection4.2canbeused.Thoughthereisonemaindierence:Forhard-decisiondecodinginsection4.2wedonotneedtoknowtheexactvalueofthelikelihoodfunction,butsimplychoosetheUetomaximizeit.Forthispurposeweonlyneedtoconcentratethefunc-tionagainsttheunknownchannel,andsimplytreatthenoisevariance2asanunknownconstantwouldbegoodenough.Howeverforsoft-decisiondecodingwehavetonotonlymaximizethelikelihoodfunctionagainstUe,buthavetoevaluatetheexactvaluesofthelikelihoodfunctionforanypossibletransmittedmatrixUe,toachievethiswehavetoconcentratethelikelihoodfunctionagainstboththeunknownchannelHandtheunknownnoisevariance2.Thelogarithmofthelikelihoodfunctionisthefollowing[16]L(YejUe;He;2)=2Nnrlog21 2NnrTr(YeHeUe)(YeHeUe)H(5.5)

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Inserting(5.5)into(5.3)givesL1(YejUe;H)=logTr(YeHeUe)(YeHeUe)H(5.6)WenextcontinuetoconcentratethelikelihoodfunctiontoeliminateHe.Maximizing(5.6)withrespecttoHe,weget^He=YeUHe(UeUHe)1=1 2YeUHe(5.7)whereweusedthefactthatUeUHe=2I.Insertionof(5.7)into(5.6)givestheconcentratedlikelihoodfunction^L(YejUe)=logTr(Ye1 2YeUHeUe)(Ye1 2YeUHeUe)H=logTrYeYHe1 2YeUHeUeYHe(5.8)Foraparticularinformationbit,says1,thelikelihoodratiocanthenbeapproximatedbyP(s1=1) PUe:s1=0expL(YejUe)=PUe:s1=11 2YeUHeUeYHeg 2YeUHeUeYHeg(5.9)Wecanthusevaluatethislikelihoodratioforeachs1,s2,s3,s4bitsforeverytimeintervalsandusethissoftinformationbitestimationastheinputofthesoft-decisionViterbidecoder.Alternatively,thereceivermayusethepreviouslyreceiveddatablockstoestimatethenoisevariance.Doingsowejustneedtoinserttheestimatednoisevarianceintothelikelihoodfunction,anditisnotnecessarytoconcentratethelikelihoodfunctionwithrespecttothenoisevariance.Ifweestimatethenoise

PAGE 55

varianceforalongenoughtimeandtaketheaveragevalue,wewillhopefullygetaveryaccurateestimation.Togettheestimated^2fromthereceiveddatablockswesimplyneedtoreferbackto(5.5)^2=1 2NnrTrn(Ye^He^Ue)(Ye^He^Ue)Ho(5.10)Substituting(5.7)into(5.10)gives^2=1 2NnrTr(Ye1 2Ye^UHe^Ue)(Ye1 2Ye^UHe^Ue)H=1 2NnrTrYeYHe1 2Ye^UHe^UeYHe(5.11)Substitutingthe^2and^He=1 2Ye^UHebackinto(5.3)yieldsthelikelihoodfunctionwithestimatednoisevarianceL(YejUe;^2)=2Nnrlog^21 ^2TrYeYHe1 2Ye^UHe^UeYHe(5.12)Withanaccurate^2wemayexpectthatthislikelihoodfunctionperformsbetterthantheconcentratedfunctionwithunknownnoisevariancesincethereisonelessunknownvariable.Figure5{2showsthesimulationresultsoftheperformanceofbothhard-decisionandsoft-decisionschemes.

PAGE 56

Figure5{2:Withrate-1/2convolutionalcoding,comparisonofBERperformanceareshownforhard-decisiondetection,soft-decisiondetectionwithconcentratedlikelihoodfunction,andsoft-decisionwithestimatednoisevariance.

PAGE 57

48

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Figure6{1:7-cellfrequencyreusesystem. duetoitssimplicity.AssumingthereexistsaLine-Of-Sight(LOS)propagationpath,thenthereceivedsignalstrengthcanbepredictedbyPr(d)=PtGtGr2

PAGE 59

Figure6{2:Typicalcelllayoutforawirelesstelecommunicationsystem. d0n(6.2)HereP0isthereceivedpoweratareferencepoint,d0isthedistancebetweenthetransmitterandthereferencepoint,andnisthepathlossexponent.Foridealfreespace,n=2.Foranurbanareawemaytaken3or4[24].Duetotheunpredictabilityoftheradioenvironmenttheabovemodelcanonlyrepresenttheaveragereceivedpower.Infacttheactualreceivedpoweructuatesvastlyaroundtheaveragevaluesincethepropagationenvironmentvariesatdierenttime/space(varyingreectiveobjects,obstructions,weather,etc).Fromasystemdesignpointofviewtheexactlevelofreceivedpowerisquitehardtopredictandthissortofpoweructuationmaybemodelledasarandomquantityinamathematicalframework.SotheLog-normalFadingmodelisbeingintroduced10log10Pr(d)[dB]=10log10P0[dB]10nlogd d0+X(6.3)

PAGE 60

whereXisazero-meanGaussiandistributedrandomvariablewithvariance2(Typical6dBoverlarge/smallscalefading).Thuswemaysaythatforanypointatthecell,theinstantaneousSNRisaGaussianrandomvariablewiththemeanvaluegivenbytheLogDistancePathLossModel.SNR= 102

PAGE 61

102

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Figure6{3:Coverageareaforthebasic/additionalmessagesasafunctionofthe\tolerable"performanceloss,fortheRb=1,Ra=1code.ThehorizontalcurvesrepresenttherelativecoverageareafortheRb=2codewithabasicmessageonly. isallowedtodecreaseto0.8timesofthatoftheoriginalcellarea,thecoverageareaoftheadditionalmessagecanonlyreach0.15timesofthatoftheoriginalcellarea,whichresultsinadecreaseinoverallsystemthroughput.Howeverforthecasen=4,theuseofanonuniformcodeincreasesthetotalthroughput.Thisobservationisnothardtounderstand;forn=4theareacloselysurroundingthebasestationenjoysarelativelymuchhigherSNRthantheareaclosetotheborderofthecellandhencethecoverageareafortheadditionalcanbeextendedfurther.Notethatthecoverageareasofthenonuniformlayeredcode(4.38)and

PAGE 63

Figure6{4:CoverageareafortheSISObasic/additionalmessagesasafunctionofthe\tolerable"performanceloss.ThehorizontalcurvesrepresenttherelativecoverageareafortheSISOrate-2codewithabasicmessageonly. (4.39)(withRb=Ra=1)tendtoapproachthatoftheRb=2code(4.46)whenincreases.Theperformanceadvantageofa2-TXsystemupon1-TXsystemisillus-tratedinFigure6{4.Thecoverageareaofthe1-TXsystemisonlyabout0:2timesasthatofa2-TXsystemforn=4.

PAGE 64

55

PAGE 65

BER,whicharealwaysloose.Wecanexpectthe\optimal"nonuniformgroupcodessuggestedinthispaperareindeednotoptimal.Futureworksmayinclude 1. CalculationofexacttheoreticalBER 2. Designingthenonuniformgroupcodesina\more"systematicalway 3. ComparetheperformanceofournonuniformSTBCtothetheoreticalcapacityboundofthebroadcastsystem

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[1] L-FWei,\Codedmodulationwithunequalerrorprobabilities,"IEEETransactionsonCommunications,vol.41,pp.1439{1449,Oct.1993. [2] ARCalderbankandNSeshadri,\Multilevelcodesforunequalerrorprotection,"IEEETransactionsonInformationTheory,vol.39,pp.1234{1248,July1993. [3] MSajadieh,FRKschischang,andALeon-Garcia,\Modulation-assistedunequalerrorprotectionoverthefadingchannel,"IEEETransactionsonVehicularTechnology,vol.47,pp.900{908,Aug.1998. [4] KRamchandran,AOrtega,KMUz,andMVetterli,\MultiresolutionbroadcastfordigitalHDTVusingjointsource/channelcoding,"IEEEJournalonSelectedAreasinCommunications,vol.11,pp.6{23,1993. [5] WLi,\OverviewofnegranularityscalabilityintheMPEG-4videostan-dard,"IEEETransactionsonCircuitsandSystemsforVideoTechnology,vol.11,pp.301{317,Mar.2001. [6] MMemarzadeh,ASabharwal,andBAazhang,\Broadcastspace-timecoding,"inProc.oftheSPIEInternationalSymposiumontheConvergenceofInformationTheoryandCommunications(ITCOM),Denver,CO,Aug.2001,pp.112{120,SPIE,Bellingham,WA. [7] C-HKuo,C-SKim,RKu,andC-CJKuo,\Embeddedspace-timecodingforwirelessbroadcast,"inProc.ofVisualCommunicationandImageProcessing,SanJose,CA,Jan.2002,pp.189{197,SPIE,Bellingham,WA. [8] MBPursleyandJMShea,\Nonuniformphase-shiftkeymodulationformultimediamulticasttransmissioninmobilewirelessnetworks,"IEEEJournalonSelectedAreasinCommunications,vol.17,pp.774{783,May1999. [9] TMCover,\Broadcastchannels,"IEEETransactionsonInformationTheory,vol.18,pp.2{14,1972. [10] PPBergmansandTMCover,\Cooperativebroadcasting,"IEEETransactionsonInformationTheory,vol.20,pp.317{324,May1974. 57

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[11] JPEG,\http://www.jpeg.org/,"JointPhotographicExpertsGroup,3/24/2003. [12] MBPursleyandJMShea,\Multimediamulticastwirelesscommunicationswithphase-shiftkeymodulationandconvolutionalcoding,"IEEEJournalonSelectedAreasinCommunications,vol.17,pp.1999{2010,1999. [13] MBPursleyandJMShea,\Adaptivenonuniformphase-shiftkeymodula-tionformultimediatracinwirelessnetworks,"IEEEJournalonSelectedAreasinCommunications,vol.18,pp.1394{1407,2000. [14] EGLarssonandPStoica,Space-TimeBlockCodingforWirelessCommuni-cations,CambridgeUniversityPress,Cambridge,UK,2003. [15] TorstenSoderstromandPetreStoica,SystemIdentication,PrenticeHallInternational,HemelHempstead,UK,1989. [16] E.G.Larsson,PStoica,andJLi,\Onmaximum-likelihooddetectionanddecodingforspace-timecodingsystems,"IEEETransactionsonSignalProcessing,vol.50,no.4,pp.937{944,Apr.2002. [17] S.M.Alamouti,\Asimpletransmitdiversitytechniqueforwirelesscommunications,"IEEEJournalonSelectedAreasinCommunications,vol.16,no.8,pp.1451{1458,Oct.1998. [18] B.L.Hughes,\Dierentialspace-timemodulation,"IEEETransactionsonInformationTheory,vol.46,no.7,pp.2567{2578,Nov.2000. [19] V.TarokhandH.Jafarkhani,\Adierentialdetectionschemefortransmitdiversity,"IEEEJournalonSelectedAreasinCommunications,vol.18,no.7,pp.1169{1174,July2000. [20] B.M.HochwaldandW.Sweldens,\Dierentialunitaryspace-timemod-ulation,"IEEETransactionsonCommunications,vol.48,no.12,pp.2041{2052,Dec.2000. [21] GGanesanandPStoica,\Dierentialmodulationusingspace-timeblockcodes,"IEEESignalProcessingLetters,vol.9,no.2,pp.57{60,Feb.2002. [22] XGXia,\Dierentialen/decodingorthogonalspace-timeblockcodeswithAPSKsignals,"IEEECommunicationsLetters,vol.6,no.4,pp.150{152,Apr.2002. [23] VTarokh,HJafarkhani,andARCalderbank,\Space-timeblockcodesfromorthogonaldesigns,"IEEETransactionsonInformationTheory,vol.45,no.5,pp.1456{1467,July1999.

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[24] TRappaport,WirelessCommunications:PrinciplesandPractice,Prentice-Hall,UpperSaddleRiver,NJ,secondedition,2002.


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NONUNIFORM SPACE-TIME CODES FOR LAYERED SOURCE CODING


By

WING HIN WONG















A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2003

































Copyright 2003

by

Wing Hin Wong
















To my family.















ACKNOWLEDGMENTS

First I have to thank my advisor, Dr. Erik G. Larsson. Without his continu-

ous guidance, this thesis could never have been a reality. I am so grateful that I

was given the opportunity to work on such a challenging topic.

Of course Dr. John M. Shea and Dr. Tan F. Wong cannot be omitted here.

I wish to express my sincere thanks to them for their supporting roles in my

committee.

The continuous support from my family is the key component to the success

of my graduate study. The gratefulness in my mind cannot be expressed by

simple languages.

Last but not least, I wish to -i thank you to all my lovely friends in

Gainesville, Florida. Their encouragement has helped make this thesis better.















TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ............................ iv

LIST OF FIGURES ................... ........ vii

ABSTRACT ...................... ........... ix

CHAPTER

1 INTRODUCTION ............................. 1

2 BROADCAST CHANNELS .......... .............. 5

3 LAYERED SOURCE CODING ......... ............. 6

4 SPACE-TIME CODING .......................... 9

4.1 Error Performance of STBC Systems ............... 9
4.2 Differential Modulation for MIMO Systems ............ 18
4.3 Nonuniform Space-Time Codes ....... .......... 21
4.3.1 Design Criteria ......... ..... ... ....... 22
4.3.2 Union Bound on the Error Probability .......... 23
4.4 Design Examples .................. .. .... .. 24
4.4.1 Rb = 1, R = 1 Code for 2 TX Antennas ... 24
4.4.2 Rb = 2, Ra = 1 Code ........... ... .. 30
4.4.3 Receive Diversity . ..... .... 30
4.5 Suboptimal Detector for Nonuniform D-STBC ... 32
4.6 Comparison of Differential Group Codes and A Differential
Alamouti Code .... ........... ..... .. 34
4.6.1 Differential Alamouti Code . .... 34
4.6.2 Alamouti Code with Differentially Encoded Symbols 36

5 CONVOLUTIONAL PRE-CODING ..... . 40

5.1 Introduction to Convolutional Codes . 40
5.2 Convolutional Codes Applied to Space-Time Coding System .42
5.2.1 Hard-Decision Decoding .................. .. 43
5.2.2 Soft-Decision Decoding .................. .. 43

6 A NETWORK APPLICATION EXAMPLE . ..... 48










7 CONCLUDING REMARKS AND FUTURE WORKS .......... 55

REFERENCES ........... ...... ......... ...... 57

BIOGRAPHICAL SKETCH ................... ...... 60















LIST OF FIGURES
Figure page

1-1 A point-to-point link. .................. .... 1

1-2 A point-to-multipoint link .............. .. 2

3-1 A nonuniform 8-PSK constellation obtained from a standard "uni-
form" QPSK constellation by splitting each original constellation
point "o" into a pair of new points "x" ... .......... 8

4-1 Received signal level may fluctuate vastly for a single radio channel. 10

4-2 Total channel gain of two independent fading channels. ...... .11

4-3 A wireless link with multi transmit and receive antennas. ....... ..12

4-4 OSTBC decouples a MIMO channel into n, number of AWGN
channels. .................. ............ ..19

4-5 Optimal values of (A, 7) for the Rb 1 R = 1 code, for a given
acceptable performance degradation of the basic message. The
results are obtained via minimization of the union bound (4.36)
and a corresponding expression for the error rate of the addi-
tional message. .................. ..... 27

4-6 Performance degradation for the basic message and performance
gain for the additional message for the Rb = R = 1 code.
The results are obtained via (4.36), along with a corresponding
expression for the additional message. The curves for the basic
message are normalized relative to the "undisturbed < i- (A
7 0) and the curves for the additional message are normalized
relative to A = 0.2, 7 0.035. ................. 28

4-7 Empirical BER for the Rb 1, Ra = 1 code with A = 0.2, 7 0.035. 29

4-8 Empirical BER for the Rb = 2, Ra = 1 nonuniform space-time
constellation. The parameters were A = 0.078, 7 = 0.018 (found
via optimization of (4.36)). .................. ... 31










4-9 Monte-Carlo Simulation of the BER performance for the subopti-
mal receiver (4.49) and (4.50) in Section 4.5, for the joint ML
receiver and for the basic message only system with Rb = 2,
R, = 1 code. The curves "- x -" and "- x -" overlap. .... 38

4-10 Monte Carlo Simulation result for the BER performance of Rb
2, Ra = 1 system, with differential group codes and differential
Alamouti codes. Note that the solid and dash curves with "x"
marks overlap. .................. ..... 39

5-1 A simple rate 1/2 convolutional encoder. . ...... 41

5-2 With rate-1/2 convolutional coding, comparison of BER perfor-
mance are shown for hard-decision detection, soft-decision detec-
tion with concentrated likelihood function, and soft-decision with
estimated noise variance. .................. .... 47

6-1 7-cell frequency reuse system. .................. .... 49

6-2 Typical cell layout for a wireless telecommunication system. 50

6-3 Coverage area for the basic/additional messages as a function of
the "tolerable" performance loss, for the Rb = R = 1 code.
The horizontal curves represent the relative coverage area for the
Rb = 2 code with a basic message only. .............. ..53

6-4 Coverage area for the SISO basic/additional messages as a function
of the "tolerable" performance loss. The horizontal curves rep-
resent the relative coverage area for the SISO rate-2 code with a
basic message only. ............... .... .. 54















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

NONUNIFORM SPACE-TIME CODES FOR LAYERED SOURCE CODING

By

Wing Hin Wong

May 2003

C('! In: Erik G. Larsson
Major Department: Electrical and Computer Engineering

We propose new space-time codes tailored to point-to-multipoint, or

broadcast, communications using 1 ,,. i t source coding. Our codes can be

encoded (and decoded) differentially, and they are based entirely on phase-shift

keying. We develop analytical design criteria for the codes, and we discuss

the design of optimal and suboptimal receiver structures. We also discuss the

relation between our new codes and a differentially encoded Alamouti code.

Numerical examples illustrate the performance of our new codes. Convolutional

codes are also introduced as a pre-coding to the space-time coding. With some

reduction of the data rate, we can significantly increase the error performance in

an adverse channel environment. Soft-decision decoding is being used to further

improve the performance of the convolutional codes. For completeness, we also

include some fundamental concepts of the space-time coding system and the

cellular system for the ease of reading for newcomers in this field.















CHAPTER 1
INTRODUCTION

During the last decade, wireless systems with multiple transmit and receive

antennas have been studied extensively, and the performance of such systems

has been proven to be extremely promising. The propagation channel associated

with a system with multiple transmit and receive antennas is sometimes called a

multiple-input multiple-output (\! M0O) channel, and the associated coding and

signal processing is referred to as space-time coding. Using proper space-time

coding, it is possible to use the degrees of freedom of the MIMO channel both

to increase the throughput and to counteract fading. For this reason, MIMO

technology is believed to become a ii' Pr cornerstone in many future wireless

communication systems.

Loosely speaking, communication links can be classified into two categories:

point-to-point links and point-to-multipoint links. In the former case, there is

exactly one transmitter and one receiver which communicate with each other at

a given time, whereas for a point-to-multipoint link, a transmitted message is

aimed at multiple different recipients simultaneously. A cellular communication

system with mobile users is an example of a point-to-point communication









Figure 1-1: A point-to-point link.






















Figure 1-2: A point-to-multipoint link.


system, whereas a radio/TV broadcast is an example of a point-to-multipoint

link (therefore point-to-multipoint links are sometimes referred to as broadcast

channels).

Point-to-multipoint links are becoming increasingly important. For in-

stance, the introduction of Digital Audio Broadcast (DAB) and High-Definition

Television (HDTV) has pioneered a whole new field of digital broadcasting

applications. As a further example, it is widely believed that much of the next

generation's wireless networking will be based on so-called ad-hoc networks,

where it may be necessary for multiple units to listen to one message at the

same time. Also, in conventional cell communication system using directional or

adaptive antennas, it is sometimes necessary to broadcast a message to the entire

cell.

There are two 1i ii differences between point-to-point and point-to-

multipoint communication links. First, a point-to-point connection can be

optimized for a given transmitter-receiver pair. For instance, a cellular system

usually employs power control techniques that adjust the transmitted power

to minimize the power consumption, reduce the amount of co-channel interfer-

ence, and at the same time ensure that the received signal strength exceeds a










certain threshold. Second, in contrast to a point-to-point communication link,

for a broadcast transmission all receivers have different capabilities to decode the

transmitted message. This is so because the different receivers experience in gen-

eral very different radio link qualities (for instance, due to varying propagation

conditions). Moreover, since quality can usually be traded for cost, the receivers

themselves may have different inherent abilities to decode the transmitted

message.

The fact that distinct receivers have different capabilities of decoding a mes-

sage si- -l. -I that the transmitted signal should consist of several components

which are of different importance for the reconstruction of the message (and

therefore have an inherently different vulnerability to transmission errors). This

idea has lead to the concept of li-, -. It source coding (e.g., [1, 2, 3, 4]), which is

now a mature technique employ, l1 in many multimedia standards. For instance,

the image coding standard JPEG-2000 and the video coding standard MPEG-4

use what is sometimes referred to as "fine granularity scalability," which enables

a gradual tradeoff between the error-free data throughput and the quality of

the reconstructed image or video sequence [5]. Such progressive source coding

methods are already in use in many Internet applications where data rate can

be traded for quality, and they are expected to pl i, an instrumental role for the

next generation of wireless standards to provide ubiquitous access both to the

Internet, and to diverse sources of streaming video and audio.

The topic of this thesis is space-time coding for broadcast channels when

1i .. I1 source coding is used. Previous works on the topic include Memarzadeh

et al. [6], which discussed "b( ,inl,,i ,iinii, techniques (hence requiring feedback of

channel state information) for a multiuser system. In Memarzadeh et al. [6], and

two kinds of beamforming techniques are discussed. Zero-forcing beamforming is










introduced which orthogonal codewords are used for each user so that inter-user

interference can be fully eliminated just as in an ideal CDMA system, provided

that exact channel state information is available at the transmitter. Another way

is the single user optimal beamformer, for which no attempt is made to avoid

inter-user interference. This is suitable for the scenario when the channel state

information at the transmitter is not very accurate, as interference suppression is

not effective.

Kuo et al. [7] proposed a 1i,. ir space-time coding scheme assuming

different receivers may have different numbers of receive antennas, and that a

receiver with more receive antennas can decode more 1 i vrs of messages.

In this thesis, we sir.-.- -I a new l t1 r, space-time coding scheme that does

not require any channel knowledge at the transmitter, and which is constructed

starting from a criterion that attempts to minimize the error rate. Our new

codes also satisfy two other important design goals. First, they can be encoded

(and detected) differentially. Second, they are entirely based on phase-shift

keying (PSK) modulation and consequently the transmitted signal has constant

envelope at all times. The codes proposed in this paper can therefore be seen

as a multidimensional extension of a transmission technique for single-antenna

systems in Pursley et al. [8].















CHAPTER 2
BROADCAST CHANNELS

Broadcast channels are defined as the simultaneous communications between

a single source and several receivers.

Early works include Cover et al. [9] and Bergmans et al. [10]. In Cover

et al. [9], They discuss the nature of broadcast channel from the perspective

of information theory, and derives upper bounds of the capacity of broadcast

channels of different types. It is proved that the maximum data rate of each

channels are jointly achievable by proper channel coding. By superimposing side

information to the common message intended for all receivers, different data rates

are possible for different channels. This idea is similar to the nonuniform codes

which we will discuss later on.

Due to the different capabilities of each receivers, and indeed the capabilities

of each receivers are varying from time to time (i.e., due to channel fading or

fluctuating noise level), the theoretical maximum capacity of a broadcast channel

is not easy to achieve in a practical system.

We have no intention to compare the performance of our nonuniform codes

with the theoretical channel capacity in this thesis, but as our nonuniform codes

contain data in multiple l V,-i we believe that we are in the right direction that

our nonuniform codes can increase the total throughput of a broadcast system.















CHAPTER 3
LAYERED SOURCE CODING

In this chapter we briefly introduce the concept of 1 i-t, I source coding.

JPEG-2000 is taken as the example here.

JPEG is an image compression standard developed by the Joint Photo-

graphic Experts Group [11]. For the encoding, the pixel bit map information of

a whole image is divided into 8 x 8 pixel blocks. Each 8 x 8 pixel block then

undergoes a discrete cosine transformation (DCT). This helps separate the image

into parts (or spectral sub-bands) of different importance (with respect to the

image's visual quality). The DCT is similar to the discrete Fourier transform as

it transforms a signal or image from the spatial domain to the frequency domain.

For most images, much of the signal energy lies at low frequencies, whereas the

higher frequency components are often small and usually they can be neglected

with little visible distortion. Thus the DCT indeed separates the original image

data into separate lv. -i~ of different importance: the low frequency components

are important for the reconstruction of the image; the higher frequency terms

are not that important although they enhance the visual quality of the image.

From this observation, we infer that the low frequency parts of data must be sent

through a reliable data channel with very good bit-error-rate (BER) performance

whereas the high frequency parts can be sent through a less reliable channel

as the occurrence of bit errors will only deteriorate the quality to a negligible

extent.










With 1 l,. i1. t source coding (for instance as mentioned above, JPEG-2000

and MPEG-4), more capable receivers, for example with higher signal-to-noise-

ratio (SNR), can achieve a higher data rate by decoding the messages contained

in all li-.-i~ while the less capable receivers can only decode the message in the

bottom 1',- -r. Thus the i, .-I important" information should be coril., -, '1 via

the bottom 1'.-r while "less impe I 1,il information can be transmitted via

higher lv,--r (s).

An example of a modulation scheme designed for 1 li, 1t 1 source coding is

nonuniform phase-shift-keying modulation (PSK), which was first introduced

by Pursley and Shea [8] (see also Pursley et al. [12, 13]). The idea is intuitively

appealing. Starting with a standard uniform PSK constellation, which contains

data for the I/'i...-" 1i- vr, a nonuniform PSK code constellation is obtained

by adding a small additional phase shift a to each original constellation point,

which contains the information for the "additional" lV-v-r. This is illustrated in

Figure 3-1, when the basic message is con,:.', i, 1 via a QPSK constellation. A

capable receiver with high SNR can distinguish among all eight constellation

points which means that both the basic and the additional 1-,-vr messages can

be decoded, whereas to a less capable receiver the constellation may appear

like a blurred QPSK constellation and hence it may only be able to distinguish

between the different "1 i;,, phase shifts; thus only the basic message can be

detected accurately. By such a construction the error rates associated with the

basic and the additional message are different. Consequently, the additional

message can carry information that is of less importance for the reconstruction

of the transmitted message than the basic message is. The error probability of

the additional message can be easily adjusted by choosing different values of the

additional phase shift a. Obviously, a larger a results in a better performance










x x


U


xTx


Figure 3-1: A nonuniform 8-PSK constellation obtained from a standard "uni-
form" QPSK constellation by splitting each original constellation point "o" into a
pair of new points "x".

for the additional message at the cost of a deterioration of the performance for
the basic message. The main virtue of this nonuniform PSK encoding scheme
is that the signaling envelope is constant while multi-i1 i-, t messages can be
transmitted with different and adjustable error performance.















CHAPTER 4
SPACE-TIME CODING

4.1 Error Performance of STBC Systems

A conventional wireless system with a single transmit and receive antenna

suffers from a large performance degradation due to the fading of the radio

channel. Fading is the fluctuation in received signal level. It is mainly due to

changes in the propagation environment, such as weather and moving reflective

objects/obstructions.

Fading can be classified as fast fading or slow fading, depending on how

rapidly the channel impulse response changes. For a fast fading channel, the

channel impulse response changes rapidly within one symbol duration, which

means that the coherence time of the channel T~ (which is a measure of the time

duration that the channel impulse response is quasi-static) is smaller than the

symbol period of the transmitted signal


T, > T (4.1)


For a slow fading channel, the channel impulse response changes very slowly

compared with one symbol duration, which means that the channel can be seen

as quasi-static for many symbol durations, that is


T, < T, (4.2)


Figure 4-1 shows an example of the fast fading of the gain of a single radio

channel. The two curves represent two channels encounter independent fast

fading. We can see that there is a high risk that a single channel may encounter










25 i

20- :




10-

5- i'

0

-5


-101-


Figure 4 1: Received signal level may


! ,,.:tuate vastly for a single radio channel.


a deep fade at a certain time. During deep fade the transmission quality may be

very poor or the transmission may simply be impossible.

To counteract the fading radio environment a multiple transmit and receive

antenna system (or MIMO) is one attractive possibility. The advantage of a

multiple transmit and receive antenna is not difficult to comprehend intuitively:

There are more than one channel which are not fully dependent, so the possi-

bility that all of them encounter a deep fading at the same time is small; thus

the deep fading problem may be solved. Figure 4-2 shows the total channel gain

(denote the individual channel gains be hi and h2, total channel gain is then

equal to ih1 2 + lhl2) of two independent fading channels. It can be seen that the

risk of the total channel gain encountering a deep fade is largely reduced.


A-







-
::
{' :iZ.,i: ".


2000


















S10 -

S5
0)

c
5' -5

-10

-15

-20 I I
0 500 1000 1500 2000
time

Figure 4 2: Total channel gain of two independent fading channels.


The corresponding encoding scheme for such a multi transmit and receive

antenna system is called space-time coding. This is an emerging topic that now

attracts many researchers' attention since its introduction about a decade ago.

The main virtue of such a multi transmit and receive antenna system is

that higher diversity order can be achieved. The diversity order can be defined

implicitly by

Bit-Error-Rate oc (SNR)-diversity (4.3)

which means that the bit-error-rate decreases as fast as the Signal-to-Noise-Ratio

(SNR) to the power of minus the diversity order. For a high SNR the advantage

of a high diversity order can be substantial.

Consider a MIMO system with nt transmit antennas, n, receive antennas

(see Figure 4-3), and assume for simplicity that the propagation channel is









TX

Y j/Y I



Y Y







nt nr

Figure 4 3: A wireless link with multi transmit and receive antennas.

linear, time-invariant and frequency flat. Let H be an n, x nt matrix whose

(m, n)th element contains the channel gain between transmit antenna n and
receive antenna m, and suppose that a code matrix X of dimension nt x N is
taken from a matrix constellation X and transmitted by sending its N columns
via the nt antennas during N time epochs. If the signal received during the same
N time intervals is arranged in an n, x N matrix Y, then

Y =HX + E (4.4)

where E is an n, x N matrix of noise. Throughout this thesis, we shall make the
somewhat standard assumption that the elements of H and E are independent
and zero-mean complex Gaussian with variances p2 and a2, respectively; hence
the channel is Rayleigh fading.
The Shannon channel capacity of such a MIMO system is a well-known
result, and it can be shown that (for details see [14] chapter 3 and the references









therein)

C(H) = B log2 II + -HPHHI (4.5)

where B is the bandwidth of the channel in Hz and P is the covariance matrix

of the sequence of transmitted data vectors


P = E[xnxH] (4.6)


If the transmitted signals are uncorrelated at different transmit antennas and we

use unit power at each transmit interval, then

1
P -I (4.7)


(4.5) reduced to

C(H) B log2 I + HHHI (4.8)
ntJ2
Clearly, the channel capacity depends on the eigenvalues of HHH.

Given Y, detection of X in a maximum-likelihood (\I ) sense amounts to

minimizing the Euclidian distance


IY HX11 (4.9)


with respect to X e X, as well as H (unless it is known). The average (over

H) probability that the ML detector mistakes a transmitted code matrix Xo

for an incorrect (and different) code matrix X / Xo is called the pairwise error

probability For completeness a brief derivation of this probability is provided here

for both coherent and noncoherent detection.

Coherent: Suppose that Xo is the true transmitted code matrix, thus

Y = HXo + E. Then the probability that an incorrect decision in favor of a








different code matrix X / X0 occurs is

P(errorIH) =P(IIY HX||2 < IIY HXo\ 2 H)

=P(IHX + El|2 < IIE 12 H)

=P(2ReTr{EHHX} > IHX 2 H) (4.0)


where X = Xo X, and I denotes the determinant of a matrix. In (4.10),
the second equality follows by some matrix algebra and the last step is a conse-
quence of the fact that 2 Re Tr{E*HX} is zero-mean Gaussian with variance
2j2 |H- HX 12 together with the ('! i 11i, bound. Under the assumptions made, the
probability density function (p.d.f.) for H is

p(H) =exp 2) (4.11)
fnrfntp2nrnt 2

Averaging (4.10) over H using (4.11), we get after some calculations

P(error) dH P(errorIH)p(H)

< (Tp2) -t dH exp ( 1-| |HX2 4H 2)
(4.12)
S(.p2) -nrt dH exp( Tr H ( XXH + I) H
p2 ~ H nI Tr < I2(X P -"rnt
= XH+ I < (X Xo)(X Xo)H I -nr
4j2 472
Noncoherent: If H is unknown, we can obtain a detection rule that depends
only on X. Consider the problem of minimizing (4.9)

min IY HX 2 (4.13)
XEX,H








15
Treating H as a deterministic unknown, we can minimize this likelihood function
with respect to H. Notice that

I|Y HX I2 y_ H XHHH 2
(xH + H )(YH XHHH) 2

= IIx(y XHHH) 2 H-(YH XHHH 2 41

+2 Re (Tr {(Y -HX)HIx HH (YH XHHH) })


nX YH XHHH 12 + H n_,y 2

Here IHxH XH(XXH)-1X is the orthogonal projector of XH. The function
in (4.14) is minimized when

HXHYH XHHH = 0

==XH(XXH)-1XyH XHHH = 0 (4.15)

tXH((XXH)-XYH HH) = 0

If X has full rank then the above leads to the result (see, e.g., [15] and [14]
appendix B)
H =YXH(XXH)-1 (4.16)

(It may be argued that if we know some sort of statistical properties of H,
we should treat H as a stochastic variable when it enters the likelihood function.
This is intuitive since we should incorporate all known information into the
detector to get the best result. But in Larsson et al. [16] (see also Larsson et al.
[14] exercise 9.7) it is shown that both approaches indeed give the same result. In
this paper we will stick to the deterministic channel approach for simplicity.)









Inserting (4.16) into (4.13) yields

min IY YXH(XXH)-1X|l2
XEX
mi mmin lYrll2 (4.17)
XEX
Smax IILxyH 112
XEX
Let


An -xf nXH (4.18)

Note that the matrix An is Hermitian, but not necessarily positive definite. Also
note that

AnXH = (fXH _HXH)XH

= X -H nHXOH (4.19)
0 0

IIxHXOH

Hence, since IIxHXoH has full rank by assumption, the matrix XolixHX0H is

positive definite.
The probability that the ML detector makes a mistake is

P(error H) P P(IXYH I2> IflXHyH 2 H)

P(Tr {(HXo +E )A(HXo +E )H} < H)

=P(2ReTr {HXoAnEH}
(4.20)
+ Tr{EAnEH} < Tr{HXoAnXHH} H)

=P(2ReTr {HXoIx HEH}

+ Tr{EAnEH} < -IlIHXoHH 2 H)
X 0


where in the last step we used the fact that Hl XlHL = nH .








17
The term EAnEH is of higher order and may be neglected in an .-i-mptotic
analysis. In particular, for a given H, the variance of its elements is of the order
a4, whereas the elements of HXollUHEH have a variance of the order a2. Since

dQ(x) 1 x2 1
S2 < -< o (4.21)
dx -2w

Thus a small finite error in variance can only caused a small finite error in the
Q-function also, so the additional small variance caused by this higher order term
can be neglected.
By the C'!I. 1 r !' bound we have

P(2ReTr{HXon HEH< -Iln XHX H 2 H

i( IHXOH 2 (4.22)
< exp (- 42 )

Hence, by averaging in a way similar to (4.23), we get

EH[P(error)] I dH P(error|H)p(H)

< (TP2 )- exp I xHH H 1 11H11
SdH e 42 H 2(4.23)
S(i~p2)-,rnt dH exp ( Tr H. -( Xo HXH + I) H

4 Xox HX o + < IXollHXo I ,4) f

To summarize, the error bounds for coherent and noncoherent detection in a
STBC system are the following

E [P(Xo X)] c < (X- Xo)(X- Xo)H -nr (42) t (4.24)
coherent 4j

and
E [P(Xo X)] < Xon Xo -n (p) (4.25)
noncoherent 4 2










A special subclass of space-time codes is the linear or/',. ',-. ','l space-time

block codes (OSTBC). The case when OSTBC applied to a 2-TX system turns

out to be the well-known Alamouti code [17].

The main virtue of OSTBC is that they achieve a diversity of order nnt for

a system with nt transmit and n, receive antennas, at an almost negligible com-

putational cost. Specifically, for such a system, a set of n, symbols {Si,..., s,}

are encoded into an nt x N matrix that has the following structure
nts
XOSTBC (A, + s B,) (4.26)
n=l

The columns of XOSTBC are transmitted via the nt transmit antennas during N

time intervals. In (4.26), {AT,B,} are matrices chosen such that


XOSTBCXOSTBC C 12. I (4.27)
n=l

for all {s,}. In general, the role of the orthogonality condition (4.27) and the

associated design of {AT, B,} is relatively well understood by now; for instance,

(4.27) implies that the symbols {s,} can be detected independently of each other

[14], so with the use of OSTBC, the MIMO channel can be decoupled into n,

number of single additive-white-gaussian-noise (AWGN) channels (See Figure

4-4).

4.2 Differential Modulation for MIMO Systems

Space-time modulation matrices that can be encoded differ. ,:l.,iall are

sometimes of special interest since such codes can easily be demodulated nonco-

herently. Differential codes with uniform error probabilities have been studied

by many authors (see, e.g., [18, 19, 20, 21, 22] for some prominent examples)

and some of them can be seen as an extension of differential PSK to MIMO

systems. In general, if t is the time index and if {U(t)} is a sequence of (square)









19














|AW

























Figure 4-4: 0' :lC d1co( .1 a MIMO0 channel into 1 number <.fl "C XWN chan-
lsignal
-------------















1el t .








information-bearing matrices, then differential encoding obtains the transmitted
code matrix X(t) at time t via

X(t) X(t )U(t) (4.28)

where X(t 1) is the matrix transmitted at time t 1. Such an encoding
usually is only meaningful under certain circumstances, for instance if U(t) is
unitary (in which case X(t) becomes unitary for all t as well, given a unitary
"initial iii i::: X (0)). By considering two (or more) received matrices Y(t)
and Y(t 1) simultaneously, noncoherent detection is possible. For example, if
we concatenate the matrices received at time t 1 and t, and assume that the
channel H remains constant over these two time intervals (to within practical
accuracy), we can write (using (4.4) and (4.28))

Y(t ) Y(t)

HX(t 1) HX(t) + E(t 1) E(t)
(4.29)
=HX(t 1) HX(t )U(t) + E(t 1) E(t)

HX(t ) I U(t) + E(t ) E(t)I

where HX(t 1) can be seen as an unknown effectivee channel matrix and

rI U(t)j is an effective code matrix. Provided that rI U(t)j is such that
the corresponding determinant in (4.25) is nonzero, noncoherent demodulation
is possible. By applying the decision rule (4.17) to the data model (4.29), the









detection rule for differential demodulation becomes
2
min (t) Y(t ) Y(t) I s]

Ymin


U(t)EX 1 U11 -
^ mmin Tr y(t 1) Y(t)
u(t)x TrY 1) Y (t) u )

*^ f ^^ f ^ ^ J ^ .^H\ ~


H(t 1)

Y(t)]

1(t) YH


a max Re Tr U(t)Y" (t)Y (t 1)
U(t)EX

where Tr{-} stands for the trace of a matrix and where we used the fact that

the information-bearing matrix U(t) is unitary. The associated average error

probability, i.e., the probability that U(O) is incorrectly detected as U for this

differential detection scheme can be bounded by using (4.25)

I --r<[rt
E [P(U( o) U )] < U ( o) H 2. ( ) ()
U()H 4(4


.31)


1 -r"
I (UU()H +Uu H
2


S2 )-nt
4(T2


For simplicity hereafter the time index t of U(t) is omitted whenever no confu-

sion can occur.

4.3 Nonuniform Space-Time Codes

Inspired by the nonuniform PSK codes discussed in Section 3, our new

nonuniform space-time codes are based on differential encoding of the product

of a unitary code matrix Ub E Xb associated with a basic message, and another

unitary code matrix Ua E X. corresponding to an additional message. Thus the

transmitted code matrix at time t is given by (cf. (4.28))


X(t) = X(t 1)Ub(tt)U t)


(4.30)


(4.32)









Using (4.31) along with the fact that UbUa is unitary, we find that the
average probability that a transmitted message pair (U ,Uo) is mistaken for
another pair (Ub,Ua) can be bounded by

E [P(Uo, U Ub, U)]

SI- (UbUaUOHUH + uuu-ru ) 2 ) nrnt (4.33)

A bound such as (4.31) or (4.33) was used in, for instance, Hughes et al. [18]
for the design of (uniform) differential space-time codes. However, although it
may be thought of as a feasible approach, an attempt to minimize this bound in
the context of r7. -,;r. :. rm space-time modulation may not produce the desired
result since the target error rates for Ub and Ua are different.
4.3.1 Design Criteria

Since the additional message can be decoded only at high SNR, the matrices

{Ua} associated with the additional message should approximately be close to
the identity matrix. Inspired by this observation, we can first consider the design

of {Ub}, treating the presence of Ua as an unmodelled noise-like disturbance
term. Doing so, we can approximately bound the error probability for Ub alone

(assuming differential detection) by

E [P(U Ub)] I- (u(0) bH + UbU0()H) -n ) 2t (4.34)2 -

where a2 is a factor that incorporates the noise-like effect of the presence of Ua.
Although the "bound" (4.34) is somewhat heuristic (and probably neither tight
nor very accurate), we believe that it may serve a purpose as a meaningful design
criterion for {Ub}.
Next, for the design of {Ua} we proceed as follows. Suppose that the SNR
is in a region such that Ub can be reliably decoded. For the design of {Ua} this









should be a reasonable assumption, since if it is not true then decoding of U. is

probably of less interest anyway. Assuming that Ub is known, the demodulation

of Us is essentially another noncoherent detection problem. To obtain a criteria

for the design of the constellation {Ua}, we want to form an error bound on U.

and average it over all possible basic messages {Ub}. Using the bound (4.25) for

noncoherent detection appears to yield criteria that are very difficult to use. As

a suboptimal approach we used instead the bound (4.24) for coherent detection.

Doing so results in the following criterion

E [P(U Us)] < (U UbUa)(Ub U UbU)O -r 2 H "
Ub EX

(U U)(U Ua)H -n. ( 2) nt

(4.35)

where in the last step Ub disappears since it is unitary. In (4.35), | denotes the

number of elements of the set.

4.3.2 Union Bound on the Error Probability

By the union bound, the probability for the basic message to be in error

(averaged over all possible pairs of additional messages) can be bounded by

E [P(basic message in error)]
1 1

(U)b ,U[))CXb (Ua Ua )EXa
k4n

I-- (U( k)U(r)U()HU b)H + U)u))U(r)HU (k)Hk -) -nt

(4.36)









Likewise, the error rate for the additional message (averaged over all possible
pairs of basic messages), can be bounded by a similar expression

E [P(additional message in error)]
1 2



I (Uk) U(U)HUn)H + U ( us) U()HUk)H -nrnt

(4.37)

These bounds will be useful for performance optimization.
4.4 Design Examples
4.4.1 Rb R = 1 Code for 2 TX Antennas
We first construct a code where the rate for the basic message is Rb 1
bit/sec/Hz and the rate for the additional message is Ra = 1 bit/sec/Hz as well.
We take the basic message Ub taken from the following set

1 0 0 -1 0 1 -1 0
SX= (4.38)
0 1 1 0 -1 0 0 -1

The constellation in (4.38), which is uniform and possesses certain optimality
properties, is due to Hughes et al. [18] and was called "BPSK" therein. Based on
the design rules in Section 4.3.1, we have handcrafted the following constellation
of matrices for the additional message Ua


( (i ear 0 e-0e 0 e-b e 0(o
Xa { } (4.39)
S0 ea 0 e"t 0 e-d 0 e-0 o

where (A, 7) are design parameters (to be discussed below).









The constellation may appear to be somewhat arbitrary but it possesses

some nice features. First, all matrices in (4.39) are unitary, this is a requirement

for MIMO differential encoding. Second, it is symmetric and uniform which

means that the error probability performance should be the same for all constel-

lation points. Also similarly to the nonuniform PSK encoding for a single channel

(cf. Section ??), multiplication with the matrices in (4.39) can be interpreted as

adding a small phase shift to the elements of the basic message Ub.

We may also think of other forms of matrices as the additional message. For

instance, consider the following anti-diagonal matrix

0 eix
Ua = (4.40)


But we immediately find that this does not work. Consider, for example,

1 0
Ub (4.41)
0 1

Then,
1 0 0 e" 0 e"^
Ub Ua = (4.42)
01 et O e07 0

It is seen that multiplying this additional message Ua not only introduces a small

phase shift but also interchanges the columns of the original basic message Ub.

This is not acceptable since our design goal is to have the additional message

only pose a small disturbance to the basic message so that for those less-capable

receivers can still detect the basic message even without knowledge of the

existence of the additional message. The U. here simply destroys the original

basic message constellation so it is not feasible.









Let us now return to our design task. If we take the initial transmit matrix,

somewhat arbitrarily


X(0) 1 (4.43)


The factor is inserted to normalize the transmitted power to unity for each

time interval. It follows that

XH(t)X(t) I= (4.44)

for all t and hence the differential encoding of the code is meaningful. Further-

more, we can verify that all elements of X have ahv--, constant magnitude:


Xk,l(t) = (4.45)

and hence all transmitted symbols are obtained by constant envelope modulation,

which was one of our design goals.

The error performance (of both the basic and the additional message) will

depend on A and 7, and typically the error performance of the additional message

can be traded against that of the basic message. The values (A,7) must be chosen

carefully. For instance, at first glance intuition would perhaps sir.-- -1 us to take

A > 0 and 7 0, but we can show that such a -wiIl,!.-" choice leads to a code

that v.1i I:-' but that does not provide maximal diversity.

In our experiments, we used the union bound (4.36) and (4.37) to optimize

over (A, 7). In particular, for a given "tolerable" loss in performance for the basic

message, we can find the pair (A, 7) that minimizes (4.37). The result is shown in

Figure 4-5. For example, if we can accept a degradation of 1.5 dB for the basic

message, then A = 0.2 and 7 = 0.035 are optimal. The optimization over (A, 7) is






























0 0.5 1 1.5 2 2.5 3 3.5 4
Maximum tolerable performance loss for basic message (dB)

Figure 4-5: Optimal values of (A, 7) for the Rb 1- R = 1 code, for a given ac-
ceptable performance degradation of the basic message. The results are obtained
via minimization of the union bound (4.36) and a corresponding expression for
the error rate of the additional message.

further illustrated in Figure 4-6, where we show how the performance associated

with the basic and additional messages varies with (A, 7).

Figure 4-7 shows the empirical bit-error-rate (BER), obtained via Monte-

Carlo simulation, for the code described above using A = 0.2, 7 = 0.035 and ML

decoding. The solid lines ("-") show the performance of differential nonuniform

BPSK for a conventional system with it = 1 transmit antenna and a single

receive antenna (this is essentially a special case of Pursley et al. [8]). The

dashed lines ("- -") show the performance for a system with nt = 2 (and a single

receive antenna) using the new code presented above. For the curves without

marks, only a basic message is transmitted. The curves with marks show the

performance when both a basic and an additional message are transmitted: the


















60-
50

40- ',I 5
0 30 0

5 20

t 10a



0.05 .05
-10
0 0.1
0.05
0 .1 5
0.15 0.2
0.2
0.25 0.25
Y

Figure 4-6: Performance degradation for the basic message and performance gain
for the additional message for the Rb = R = 1 code. The results are obtained
via (4.36), along with a corresponding expression for the additional message. The
curves for the basic message are normalized relative to the "undisturbed < i-
(A = 7 = 0) and the curves for the additional message are normalized relative to
A 0.2, 7 0.035.


curves marked with "o" show the BER for the basic message, and the curves

with "x" show the BER for the additional message. Clearly, the transmit

diversity system outperforms the conventional one -observe, in particular, the

different slopes of the BER curves both for the basic and for the additional

message. The simulation also confirms that the transmission of an additional

message incurs a small performance degradation for the basic message.








29








10 -
nt=l, DBPSK (basic msg. alone)
-- nt=1, DBPSK (basic msg.)
nt=1, DBPSK (add. msg.)
n2 t=2, Code (4.38) (basic msg. alone)
10 nt=2, Code (4.38)-(4.39) (basic msg.)
(, -,- nt=2, Code (4.38)-(4.39) (add. msg.)


$10 S -
I t

m *%


10-4




10-5
15 20 25 30 35
Signal-to-noise ratio [dB]

Figure 4-7: Empirical BER for the Rb 1, R = 1 code with A 0.2, 7 0.035.









4.4.2 Rb = 2, R = 1 Code

To obtain a code with a higher information rate for the basic message, we

next take Ub from the following algebraic group of 16 matrices (which is due to

Hughes et al. [18] as well) generated by

( ej,/4 0 0 -1 \
b =( (4.46)
0 e-j-/4 1 0

The notation in (4.46) means that all possible matrices can be generated by

choosing any arbitrary integers M and N in the following expression
M -N
ejw/4 0 0 -1
(4.47)
0 e-j-/4 1 0

The constellation used for the additional message is chosen to be the same as

(4.39), so we now have a system with rate Rb = 2 for the basic message and rate

Ra = 1 for the additional message. The corresponding simulated BER is shown in

Figure 4-8.

4.4.3 Receive Diversity

To demonstrate the transmit diversity achieved for the 2-TX system in the

simplest way, our results above are based on the 1-RX case. Although of this, our

newly proposed nonuniform STBC codes are not only useful for 1-RX case but

also for a general n,-RX case. This is obvious since our theoretical derivations

above are not tailored for the 1-RX case, hence it should work automatically

for a general n,-RX case. Also we could expect a diversity order of nnt can be

achieved. This is so since we can see the data model of a n,-RX nt-TX system as













**%
* K


10-1





10-2
-2-



( -3
10
Il




0
I,

m
10-3


% I


30
Signal-to-noise ratio [dB]


Figure 4-8: Empirical BER for the Rb = 2, Ra
stellation. The parameters were A = 0.078, 7 =
(4.36)).


n, different 1-RX nt-TX system

Y =HX+ E

Y,





Y,




Yn


= 1 nonuniform space-time con-
0.018 (found via optimization of


(4.48)


H, X + En,


4Q
44k
4,


nt=l, DQPSK (basic msg. alone)
- nt=1, DQPSK (basic msg.)
- nt=1, BPSK (add. msg.)
nt=2, Code (4.46) (basic msg. alone)
O nt=2, Code (4.46)-(4.39) (basic msg.)
-. nt=2, Code (4.46)-(4.39) (add. msg.)


H1 El

X +E


HX+E E,
HIX + E,


0


10-5
2(









where Yk for k =1,. n, is the 1 x N received data at the kth receive antenna,

Hk is the 1 x nt channel matrix associated with the kth receive antenna, Ek is

the 1 x nt noise matrix at the kth receive antenna.

As usual, for simplicity if we assume the channel matrices associated with

n, different receive antennas Hk (for k = 1, n,) are independent and the

noise is white (El, E, are uncorrelated), then the n,-RX nt-TX can be

broken down into n, number of independent 1-RX nt-TX system. Each receive

antenna has sufficient data to detect the transmitted matrix X so they can work

individually and achieve transmit diversity of order nt. To further utilize receive

diversity we can employ joint ML detection across all n, independent receive

antennas thus increasing the overall diversity by a factor of nr. Since the n,

receive antennas are independent at the receiver site, so optimizing the space-

time encoding scheme for a single receive antenna is equivalent to optimizing a

system with n, independent receive antennas.

4.5 Suboptimal Detector for Nonuniform D-STBC

To detect both the basic and the additional messages in a joint ML fashion

we have to search through all IXb IX. possible combinations to find the

combination of Ub and Us which maximize the cost function in (4.30). This may

be computationally burdensome if the constellation sizes of both lwr-is are large.

A possible remedy to this problem is to use a suboptimal detection rule described

as follows

Step 1. Detect the basic message only using (4.17), neglecting the presence

of the additional message
r 2
ub (t)= b in Y(t Y(t)] ub(t)
tUb(tEb L (4.49)

Smax Re (Tr Ub(t)YH(t)y(t )}
Ub (t)EXb









Step 2. Detect the additional message assuming the decision taken in Step 1
is correct
r 2
mm Y(t ( U t) U ) Y t) UH
U 1(4.50)
m> ax Re Tr U (t)YH Y l)
Ua(t)EXa L I
Let us elaborate more on why this suboptimal detection scheme works. Since

Ua is close to the identity matrix, when compared to the basic message Ub it can
be treated as small unmodelled noise; hence the detection rule in Step 1 can still
be able to detect the basic message with only a small performance degradation.
As the error performance of the basic message is typically superior to that of the
additional message, we can assume the detected basic message is correct when we
are dealing with the detection of the additional message. Thus the observation
model for the detection of Ua(t) becomes

Y(t -1) rb(t) Y(t)]

=HX(t b() Ub(t)Ua(t) + E(t ) E(t) (4.51)

HX(t 1)b(t) I Ua(t)] + E(t 1) E(t)

Treating HX(t 1)Ub(t) as an unknown effectt, channel as in (4.29) and
using (4.17) and (4.30) yields the detection rule in Step 2.

By applying this suboptimal detection rule the computational complexity
is reduced from IXb IX | to IXb + |11. For example, for the R = 2, Ra = 1
code, the computational time is reduced from XbI, = 16 4 = 64 to

IXb + I1 = 16 + 4 = 20. For this example the reduction is not very significant;
yet this suboptimal detection scheme does provide more flexibility as opposed









to the joint ML detection because if only the basic message is needed, the

computational load will be further reduced to XbI = 16 (for this example).

We expect that there is some performance loss associated with this subopti-

mal detection scheme since it does not exploit knowledge of the structure of the

additional message in the Step 1.

The Monte-Carlo simulation result in Figure 4-9 shows the performance

of the suboptimal detector for the Rb = 2, Ra = 1 code compared to the

performance of the joint ML detector. The detection of the basic message gives

a small performance loss (a 0.3 dB) compared to the joint ML detector, whereas

the detection of the additional message is (to within practical accuracy) equal to

that of the joint ML detector.

4.6 Comparison of Differential Group Codes and A Differential Alamouti Code

4.6.1 Differential Alamouti Code

The main advantage of the nonuniform space-time group codes introduced in

Section 4.4 is that the signal envelope is constant. However, also for this reason

they cannot achieve the maximum possible performance (in terms of BER)

because they do not make use of any amplitude information in the signal. On

the contrary, orthogonal space-time block codes (OSTBC) should provide better

performance (BER) since they are already optimized by construction [17, 23, 14].

For 2 TX antennas OSTBC reduces to Alamouti code. The construction of an

Alamouti code matrix is as follows


U(t) -( ) 2 (4.52)


where si and s2 are the information bearing symbols and (.)* stands for the

complex conjugate. For the Rb = 1 system, si and s2 are taken from the BPSK

set {-1, 1}, and for the Rb = 2 system, they are taken from the QPSK set









{-1, -, ,j}. The 1//2 factor is included for normalization purposes so that

the transmitted power per time interval equals one. This Alamouti matrix U(t)

can then be encoded differentially by using (4.28) in Section 4.2. Note that the

Alamouti code works together with the additional message U. taken from (4.39)

to form a nonuniform constellation, and differentially encoded by (4.32), but

unfortunately the matrices in the so-obtained constellation do not have constant

signal envelope.

Here we compare the performance of the group codes to that of the Alam-

outi code. The Rb = 1 group codes (4.38) in Section 4.4 and Rb = 1 Alamouti

codes are indeed equivalent, so there is no performance advantage for Alamouti

codes for this case. The equivalence of both codes is easy to show as in this case

the set of Alamouti matrices become

1 1 1 1 1 -1 1 -1 1 1 -1 -1
Xa amout 7 { [ j (4.53)
V 1 -1 -1 -1 1 / -1 1_ t


If we pre-multiply the above set by ] then we obtain the Rb 1 group

code in (4.38).

For the Rb = 2 codes, the Alamouti code does provide somewhat better

BER performance than the group codes in (4.46) and (4.39). This is so since

the Alamouti code uses both signal amplitude and phase to transmit informa-

tion; thus it should be less susceptible to the phase disturbance caused by the

additional message. Figure 4-10 shows the simulated performance difference

between the Rb = 2, Ra = 1 nonuniform group codes in (4.46) and (4.39) and

a Rb = 2 Alamouti code together with the additional message (4.39). For the

basic message the Alamouti code outperforms the group code by about 1dB, and

it is somewhat more robust when additional message is added. Meanwhile the









performances of the detection of the additional message are equal both when the
Alamouti and the group codes (4.39) are used as the basic message carrier.
4.6.2 Alamouti Code with Differentially Encoded Symbols
It may be argued that both the constant signaling envelope and maximum
BER performance can be achieved at the same time by using symbol-wise
differential encoding and forming the Alamouti matrix afterwards. However,
unfortunately this encoding scheme does not work. This is demonstrated as
follows.
Suppose that we employ symbol-wise differential encoding


(4.54)
x2(t) x2(t- 1) s(t)

and use these elements to form the Alamouti matrix

X(t 1) xi(t-1 1) x (t-1)
X(t t) =
X2(t -) -Xt(t -)
J (4.55)
X( x(t) x) (t) xI(t- 1)s(t) xt-(t 1)s(t)
X2(t) -X*t) x2t-l)St) -x*I(t l)S*Lt)

The above code matrix has the property that Ixi(t)| = x2(t)l = I(t)l 2(t)
1 for all t and hence the signal envelope is constant for both transmit antennas
all the time. Also X(t) is unitary for all t since it is an Alamouti matrix. The
observation model (4.29) becomes (cf. Section 4.2)


rY(t 1) Y(t)
=H X(t1) X(t) + E(t 1) E(t)] (4.56)

HX(t ) I XH(t i)X(t) + E(t 1) E(t)









Employing the noncoherent detection rule (4.30 to the above model (4.56) the

following detection rule is obtained


max Re{ Tr. (XH(t 1)X(t)Y-(t)Y (t -))j I
si(t),s2(t)eS I I
= max
s1 (t),s2 (t)CS
R (t 1) ( 1) ) st (t) -(t 1) s (t) (t) 1
2(t) -X(t-1) 1) [2(t 1) s2(t) -X (t- 1). s(t)
Smax
s (t),s2(t)e
Re Tr t 2 (t) + 2 (t) z(t l)x) (t 1) (t) S;(t))
X1 [ _( (t l)X2 (t 1) (s1(t) s2(t)) *(t)+ S* (t)


1)}}
(4.57)


The above maximization problem (4.57) cannot be solved unless xi(t 1) and

x2(t 1) are known. For example, consider the following two sets of values


{sI(t),S 2(t),t 1)X2(t 1)} {1, -1,1}


(4.58)


{1(t), S2(,x(t 1)X2(t 1)} {-1,1, -1}


(4.59)


These make the metric in (4.57) equal to


Re Tr 0 -2 YH(tY(t 1) I
2 0

which implies that the sets {si, a2} = {1, -1} and {si, 2} =

identifiable if xl(t 1) and x2(t 1) are unknown.


(4.60)


{-1, 1 are not


and








38







10 -1
nt=2, Code (4.46) (basic msg. alone)
-- nt=2, Suboptimal detector, Code (4.46)-(4.39) (basic msg.)
-- nt=2, Suboptimal detector, Code (4.46)-(4.39) (add. msg.)

10-2 nt=2, Joint-ML, Code (4.46)-(4.39) (basic msg.)
-. nt=2, Joint-ML, Code (4.46)-(4.39) (add. msg.)




I
I 2



10-4






15 20 25 30 35
Signal-to-noise ratio [dB]

Figure 4-9: Monte-Carlo Simulation of the BER performance for the suboptimal
receiver (4.49) and (4.50) in Section 4.5, for the joint ML receiver and for the
basic message only system with Rb = 2, R = 1 code. The curves "- x -" and
"- x -" overlap.
















10-1
10 II
nt=2, Group Codes (4.46) (basic msg. alone)
-' nt=2, Group Codes (4.46)-(4.39) (basic msg.)
-. nt=2, Groups Codes (4.46)-(4.39) (add. msg.)
Sn=2, Alamouti Codes (4.52) (basic msg. alone)
*1 n t2, Alamouti Codes (4.52)-(4.39) (basic msg.)
"1 nt=2, Alamouti Codes (4.52)-(4.39) (add. msg.)

a -
10 -

a )






I I
10-4





15 20 25 30 35 40
Signal-to-noise ratio [dB]

Figure 4-10: Monte Carlo Simulation result for the BER performance of R, = 2,
Ra = 1 system, with differential group codes and differential Alamouti codes.
Note that the solid and dash curves with "x" marks overlap.















CHAPTER 5
CONVOLUTIONAL PRE-CODING

5.1 Introduction to Convolutional Codes

In Pursley et al. [12], convolutional coding are introduced for extra error

protection before the data symbols are further encoded by the nonuniform

PSK modulation. For completeness we introduce here the basic concept of

convolutional coding as a tool to improve the BER performance.

Convolutional coding is a kind of forward error correction (FEC) technique,

which means that no feedback channel is required in contrast to, e.g. Automatic

Repeat Request (ARQ).

The idea of convolutional coding is simple: To improve the error perfor-

mance of a transmission link by adding some carefully designed redundant bits

to the data before it is transmitted through the channel. This process of adding

redundancy is known as channel coding. In contrast to block codes which operate

on large message blocks, convolutional codes operate on a continuous stream of

data bits.

Convolutional coding is a FEC technique which is especially suitable to

channels which are contaminated by additive white gaussian noise (AWGN). By

using convolutional coding, BER performance can be improved significantly at

the same SNR, at the cost of lowering the data rate. Meanwhile this rate loss can

be compensated for by increasing the constellation size.

There are two important parameters for a particular convolutional codes,

namely the code rate and the constraint length. The code rate, k/n, is simply the

ratio of the number of data bits input to the convolutional encoder (k) to the










First output







Input ZA-----t--- -- Z-1




\ //





Figure 5 1: A simple rate 1/2 convolutional encoder.


number of output bits by the convolutional encoder (n), can also be a measure

of the efficiency of the code. The constraint length, K, denotes the "1. 1,;I !: or

"memory" of the convolutional encoder, that is, each output bit is affected by K

previous input bits.

For years, convolutional coding has been the predominant FEC technique.

But there is a tradeoff of employing convolutional coding. For example, using

a 1/2 rate convolutional codes, the data rate will be deducted to half, provided

that the modulation technique is the same. This is simply because with rate 1/2

convolutional encoding, two channel symbols are needed to be transmitted per

one data bit. But the advantage is that, the convolutional coding can improve

the error performance at the same SNR, at the cost that the data rate will

decrease by a factor of n/k.

Figure 5 1 shows a typical rate 1/2 convolutional encoder, built with two

shift registers and two modulo-2 adders.










One should p ,i' attention that an interleaver is usually used to interleave

the encoded data bit stream from the output of the convolutional encoder.

The benefit of interleaving the data bits is that it can provide time diversity:

consecutive bits are likely to be contaminated by a temporary strong interference

or noise, or sustained by deep fading channel environment, this can make the

error correcting algorithm at the convolutional decoder fails to correct bit errors

and recover the correct information. With sufficient interleaving, the originally

consecutive bits are now separated far apart as they are being transmitted

through the wireless channel, so that we can reasonably assume that the bits

nearby each other after de-interleaving have undergone independent fading

and are affected by independent noise and interference, this is desirable to the

convolutional decoder as consecutive bits are not likely to encounter errors

simultaneously. Also the design task of the ML detector at the receiver is also

simplified.

5.2 Convolutional Codes Applied to Space-Time Coding System

We are going to investigate the error performance of the system if a con-

volutional encoder is employ, -l to the information bit streams prior to STBC

mapping. It is expected that the BER performance can be improved substan-

tially by such kind of pre-coding in an adverse channel environment. This simply

trades the error performance with data rate.

To start with, we choose a simple rate-1/2 convolutional encoder with

constraint length K = 3, generators are 5, 7 in octal.

As for our Rb = ,R, 1 code, there are two bits carried in the basic 1liv.

and two bits carried in the additional 1-~v r for each transmit time interval, let

sl, s2 denote the two bits carried in the basic 1-v. r and S3, s4 denote the twos

bits carried in the additional 1 -r, respectively. And now we have four separate










(independent) bit streams, for example they may look like the following

i 0, 1, 1,0, 1, 1,0, 1, 1, *

2 1, 0, 1, 1 0, 0,0, 1,
(5.1)
s3 = 0,0,0, 1,0, 1, 1, 1, 1,

S4 = 0, 0, 1,0, 0, 1, 0, 1,0,...

We model the sl, 82, 83, 84 as independent bit streams with random 0 and 1. For

each bit stream we pass it through a stand-alone binary convolutional encoder

stated above. After that, we would have four encoded bit streams, denoted by

s s2, s3, s. Notice that the encoded streams are not of the length of the raw

data streams, for the rate-1/2 encoder used in our study here, the length of the

encoded streams is doubled, we sacrifice the data rate by half to trade for better

BER performance.

The four encoded bit streams are then being passed through the STBC

mapper as before and sent via the MIMO system.

5.2.1 Hard-Decision Decoding

We get the estimated bit streams si', S2, S3', 54' at the output of the

hard-decision symbol detector, we then pass them through the Viterbi decoder

to decode them back into the raw data streams. BER performance of such a

hard-decision decoding system is shown comparing to the un-encoded system.

5.2.2 Soft-Decision Decoding

For a soft-decision decoder, the input is no longer a precise 0 or 1 binary

estimation but the likelihood ratio P(c = 1)/P(c = 0) of each bits.

To calculate the likelihood ratio of each individual bit we have to consider

the receiver structure. For the differential encoding and decoding system in

section 4.2, to decode the differentially encoded information message, the receiver










has to consider two consecutive received data blocks


Y(t 1) Y(t) = HX(t ) I U(t) + E(t 1) E(t) (5.2)
S------ effective channel He '' v-
Ye Ue Ee

where Ye and Ue stand for the "effectli, received data block and the trans-

mitted matrix, respectively. To detect the transmitted matrix Ue given Y,, the

noncoherent detection technique discussed in section 4.2 can be used. Though

there is one main difference: For hard-decision decoding in section 4.2 we do

not need to know the exact value of the likelihood function, but simply choose

the Ue to maximize it. For this purpose we only need to concentrate the func-

tion against the unknown channel, and simply treat the noise variance a2 as an

unknown constant would be good enough.

However for soft-decision decoding we have to not only maximize the

likelihood function against U,, but have to evaluate the exact values of the

likelihood function for any possible transmitted matrix Ue, to achieve this we

have to concentrate the likelihood function against both the unknown channel H

and the unknown noise variance a2. The logarithm of the likelihood function is

the following [16]


L(YIU,, He, a2)= -2Nnr log a2 -_ 2Tr ((Y HeUe)(Y HeUe)H} (5.3)


Maximization of (5.3) with respect to a2 requires that

9L(Y I U, H, a2) 0
a72 (5.4)
1 1
S-2Nnr- + -Tr {(Y HeUe)(Y HeUe)H} 0
a2 a4

which yields a positive solution for a2


a2 = tTr ((Y HU,)(Ye HU,)H} (5.5)
2Nn,









Inserting (5.5) into (5.3) gives


L1(Y U, H) = log Tr {(Y H,U)(Y, HU,)H} (5.6)

We next continue to concentrate the likelihood function to eliminate He.

Maximizing (5.6) with respect to He, we get

He YUH(U H)-U 1 tYUH (5.7)
2

where we used the fact that UUH = 21.

Insertion of (5.7) into (5.6) gives the concentrated likelihood function



L(YIU,) -logTr (Y,- -YuUU) (e eUH )H
(5.8)
-logTr YYY HYe eH eYH

For a particular information bit, -, sl, the likelihood ratio can then be

approximated by

P(si 1) UZ,:I exp (L(Ye Ue))
P(si 0) ugs:, oexp (L(Y, U ))
(5.9)
S:Ucs81 Tr{yyY, Uyu0uyY}
1
1Ugsi 0 Tr{ Y0YH Y0Iuuy}

We can thus evaluate this likelihood ratio for each sl, s2, 83, S4 bits for every

time intervals and use this soft information bit estimation as the input of the

soft-decision Viterbi decoder.

Alternatively, the receiver may use the previously received data blocks to

estimate the noise variance. Doing so we just need to insert the estimated noise

variance into the likelihood function, and it is not necessary to concentrate the

likelihood function with respect to the noise variance. If we estimate the noise









variance for a long enough time and take the average value, we will hopefully

get a very accurate estimation. To get the estimated P2 from the received data

blocks we simply need to refer back to (5.5)

62 = Tr Ye H,U,)(Y, -E H,U,)H} (5.10)
2Nnr

Substituting (5.7) into (5.10) gives

1 1 H 1 H
2 Tr (Y YIU UU)(Y _YU U) H
2Nnr 2 2 (5.)
1 r H H(5.11)
1 Tr YYH y -Y_ U U T \Y
2Nn, 2ee
H
Substituting the 62 and He = YUU back into (5.3) yields the likelihood

function with estimated noise variance
2) Tr l _tH t-Y'U( H( vy
L(YIU,,2) --2Nrnlog 2 -Tr 1Y -H UY (5.12)

With an accurate c2 we may expect that this likelihood function performs

better than the concentrated function with unknown noise variance since there

is one less unknown variable. Figure 5-2 shows the simulation results of the

performance of both hard-decision and soft-decision schemes.






















U ^

0%

8 .
\ ^
v "-

\\5
\\
\\
%%, Y3
%.%
%


SBasic msg. (hard decision)
Add. msg. (hard decision)
) Basic msg. (soft decision with concentrated ML function)
Add. msg. (soft decision with concentrated M L function)
Basic msg. (soft decision with exactly known noise variance)
Add. msg. (soft decision with exactly known noise variance)
)Basic msg. (soft decision with est. noise variance)
Add. msg. (soft decision with est. noise variance)
-, S --
X\
'Xx


100




10-1




10-2
I
O -2



0
10-3
I


-
10-4


0

111111 ,


8 10 12 14
Signal-to-noise ratio [dB]


16 18


Figure 5-2: With rate-1/2 convolutional coding, comparison of BER performance
are shown for hard-decision detection, soft-decision detection with concentrated
likelihood function, and soft-decision with estimated noise variance.


'.0~
C>


10-5
4


I


\N
\\















CHAPTER 6
A NETWORK APPLICATION EXAMPLE

We consider employing our nonuniform space-time codes in a broadcasting

telecommunication system. First we present a short introduction to contempo-

rary typical wireless cellular systems.

The concept of a wireless cellular system is fairly simple: a large region to be

served are divided into many small areas called "cells". Each cell is serviced by

one or more base stations) located at the center or the corners of the cell. The

advantage of such a cellular system is that the available frequencies band can be

reused in cells which are far apart; thus the same available frequency band can

now be enjoi, l by a small area which may consist of several small cells.

Figure 6-1 shows a 7-cell frequency reuse scheme. A cluster is formed

by seven cells. A cluster can use all the available bandwidth, and the whole

frequency band is being reused in all other clusters. So the whole frequency band

is now shared by seven cells only, thus each user in the cluster should be able

to get a adequate amount of usable bandwidth. Since the .,.i i:ent clusters use

exactly the same frequency band, a power control scheme can be employ, l to

keep the Signal-to-Interference-Ratio (SIR) under certain threshold.

Now we are moving on to see what is happening on one particular cell.

Figure 6-2 shows a typical cell. For simplicity we assume the shape of the cell to

be circular, and that the base station is located at the center. Let Ro represents

the cell radius. Typically the SNR at the cell border should be the lowest. To

predict the signal strength at various places of the cell, some sort of propagation

model has to be used. Here we first introduce the free space propagation model




























Figure 6 1: 7-cell frequency reuse system.


due to its simplicity. Assuming there exists a Line-Of-Sight (LOS) propagation

path, then the received signal strength can be predicted by

PtGtG,A 2
Pr(d) (= 4 2 (6.1)
(4w)2d2

where Pt is the transmit power at the transmit antenna, Gt and G, are the

transmit and receive antenna gains respectively, A is the wavelength of the

transmitted signal, d is the propagation distance. In the previous discussion of

space-time coding it is assumed that we have a frequ, -ii'- -flat channel, this means

that we must use a narrowband modulated signal.

It is well known that the received signal strength attenuates as d-2 in free

space theoretically, but in practical environment for a wireless link, the LOS

path usually not exists and the propagation may encounter various loss such as

scattering and other obstructions, so the attenuation is usually much higher.

Even with a LOS path, the received signal strength usually attenuates faster

than d-2 due to destructive reflection. Another commonly used model is the




















Ra RbRo




Figure 6-2: Typical cell layout for a wireless telecommunication system.

i, -./:, ir,.. : path loss model


P (d) = Po d (6.2)
do

Here Po is the received power at a reference point, do is the distance between the

transmitter and the reference point, and n is the path loss exponent. For ideal

free space, n = 2. For an urban area we may take n 3 or 4 [24].

Due to the unpredictability of the radio environment the above model can

only represent the average received power. In fact the actual received power

fluctuates vastly around the average value since the propagation environment

varies at different time/space (varying reflective objects, obstructions, weather,

etc). From a system design point of view the exact level of received power is

quite hard to predict and this sort of power fluctuation may be modelled as a

random quantity in a mathematical framework. So the Log-normal Fading model

is being introduced


lOloglo rP(d) [dB] = lOloglo Po [dB]- 10 log( ) + X, (6.3)
do









51
where X, is a zero-mean Gaussian distributed random variable with variance O2

(Typical w 6 dB over large/small scale fading).

Thus we may w that for any point at the cell, the instantaneous SNR is a

Gaussian random variable with the mean value given by the Log Distance Path

Loss Model.

SNR SNR + X, (6.4)

Since now the SNR at any point is modelled as a random variable then the

instantaneous BER performance is also random, we can define the coverage area

of a cell.

Define the coverage radius to be the radius of the area such that a specified

detection performance (in terms of BER) is achievable with a certain probability

that we call the Quality of Service (QoS). For example, if we let Ro denote

the 9' coverage radius then within the area of radius Ro the message can be

detected at a specified BER during 9' '. the of total time.

If nonuniform space-time codes are used to increase the data rate for the

more capable receivers, it is clear that the coverage radius for the transmission of

the basic message under the same QoS will decrease to Rb < R0. Meanwhile the

coverage radius of the transmission of the additional message now becomes Ra;

of course its value depends on QoS for the additional message. For the simplicity

of our analysis here let us assume the same QoS for both the basic and the

additional message.

Denote the "tolerable" performance loss for the detection of the basic

message as A dB. Assuming log-distance fading it can be shown that


R 0-o (6.5)
Ro








52
R2 r2
R 10- Ti (6.6)
-"o
where F is the performance difference (in dB) between the detection of the

additional message and that of the standard uniform constellation system. The

parameter n is the path loss exponent for the radio link environment (i.e. Power

oc (distance)-"; for details see Rappaport et al. [24]). The above equations

are also valid in a log-normal fading model. It is not difficult to obtain the

result above: Denote the SNR required to have a certain BER performance as

SNRth[dB]. Then, for example, the 9'-'. coverage radius Ro can be determined

as it satisfies

P [Pr\(Ro)[dB] + Xsigma > SNRth[dB] 9= (6.7)

When we apply the nonuniform modulation the BER performance of the basic

message deteriorates by A dB, which means that it requires a A dB higher SNR

to obtain the same performance, so


P P(Rb)dB] + Xsigma > SNRth[dB] + A] = 9' I (6.8)

Comparing (6.7) and (6.8) we get

Pr(Rb)[dB] P,(Ro)[dB] + A
R. ((6.9)
tlOloglo Po [dB] 10n log( ) lOlog Po [dB] + A(6.9)
RO

and (6.5) follows immediately, (6.6) can be shown similarly.

Figure 6-3 shows how Rb and Ra vary with A for the Rb = Ra 1

system, with the power attenuation factor of the radio link taken to be n = 2 and

n = 4; these are typical values for free-space and downtown areas, respectively

[24]. It can be seen that employing the nonuniform constellation does not

increase the total throughput of the system (i.e., the sum of the data rate for all

users) in the case n = 2. For example, if the coverage area for the basic message









53
(0 1 Illl--- ] ------ |i ii l
--e- Basic msg. (n=2)
=0.9e -N-- Add. msg. (n=2)
o R =2 basic msg. only (n=2)

S0.8 -o- Basic msg. (n=4)
S, -x- Add. msg. (n=4)
S_ R =2 basic msg. only (n=4)
o 0.7 b

0.6
----------------- --- -^------------

) 0.5-
N

E 0.4- -x- -
o ***
0 X*
-0.3- -

c0.2 -

S0.1

0 I
0 0.5 1 1.5 2 2.5 3 3.5 4
Performance loss of basic msg. (dB)

Figure 6-3: Coverage area for the basic/additional messages as a function of the
"tolerable" performance loss, for the Rb 1, R = 1 code. The horizontal curves
represent the relative coverage area for the Rb = 2 code with a basic message
only.


is allowed to decrease to 0.8 times of that of the original cell area, the coverage

area of the additional message can only reach 0.15 times of that of the original

cell area, which results in a decrease in overall system throughput. However for

the case n = 4, the use of a nonuniform code increases the total throughput. This

observation is not hard to understand; for n = 4 the area closely surrounding

the base station enjoys a relatively much higher SNR than the area close to the

border of the cell and hence the coverage area for the additional can be extended

further. Note that the coverage areas of the nonuniform lF ,v. 1, code (4.38) and














0.


0.


En






o
Z0


cu
a),




0
0(
O II"
8 C

^I


0.


-e- SISO basic msg. (n=2)
45 --- SISO add. msg. (n=2)
Rate-2 SISO basic msg. only (n=2)
-o- SISO basic msg. (n=4)
-- SISO add. msg. (n=4)
Rate-2 SISO basic msg. only (n=4)
35 -




25-
-------------- --------------

-Jc-
25 -~--~ -
.2- .-
lb --

M--




05- -
-n1-

......,_.--------


0 0.5 1 1.5 2 2
Performance loss of SISO basic msg. (dB)


Figure 6-4: Coverage area for the SISO basic/additional messages as a function
of the "tolerable" performance loss. The horizontal curves represent the relative
coverage area for the SISO rate-2 code with a basic message only.


(4.39) (with Rb Ra = 1) tend to approach that of the Rb = 2 code (4.46) when

A increases.

The performance advantage of a 2-TX system upon 1-TX system is illus-

trated in Figure 6-4. The coverage area of the 1-TX system is only about 0.2


times as that of a 2-TX system for n = 4.


0.


0


0


0















CHAPTER 7
CONCLUDING REMARKS AND FUTURE WORKS

We have presented new nonuniform space-time codes that can be encoded

and detected differentially, and that are based entirely on phase-shift keying.

We also discussed analytical criteria for code construction and optimization,

and we compared its performance with that of a scheme based on the Alamouti

code. We also studied a suboptimal detector and its performance, which we

found to be satisfactory (a loss within 0.3 dB compared to joint ML decoding).

We also demonstrated how nonuniform space-time codes in a broadcasting

telecommunication system can increase the total throughput.

It may be argued that using already established nonuniform constellations

for single transmit antenna systems together with, for instance, known linear

space-time codes should be a natural approach to the problem of designing

nonuniform space-time constellations. However, there are at least two problems

associated with such an approach. First, it may in general not be optimal,

simply because we are optimizing over the class of space-time codes and the class

of nonuniform single-antenna constellations separately, instead of optimizing over

the class of nonuniform MIMO constellations. Second, we found it difficult to

incorporate constraints (such as constant envelope after differential encoding),

that are desired or required from a practical implementation point of view.

Therefore, we believe that it may be advantageous to design nonuniform space-

time constellations by approaching the problem from first principles.

In this thesis, the nonuniform space-time block codes are chosen in quite

an ,il I trary" way, and they are optimized with the help of chernoff bounds of










BER, which are ahv--, loose. We can expect the ..p 'l ii',!" nonuniform group

codes -, ii-. -1. in this paper are indeed not optimal. Future works may include

1. Calculation of exact theoretical BER

2. Designing the nonuniform group codes in a "more" systematical way

3. Compare the performance of our nonuniform STBC to the theoretical

capacity bound of the broadcast system















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59
[241 T Rappaport, Wireless Communications: Principles and Practice, Prentice-
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BIOGRAPHICAL SKETCH

Wing Hin Wong was born on T ,- 26, 1977, in Hong Kong. He received

his bachelor's degree in mechanical engineering with first class honors, from the

University of Hong Kong, in 1999. Since August 2001, he has been pursuing a

Master of Science Degree in electrical and computer engineering at the University

of Florida in the area of wireless communications.